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+{"text":"\\section{INTRODUCTION}\nLiquid helium ($^4$He) has a reputation for being the first substance in which\none is able to observe many macroscopic quantum phenomena. In particular, it\nwas the first system that could sustain superfluid flow, \\cite{bulk} and as a\nconsequence display a number of amazing properties such as second sound,\nquantized vortices and the fountain effect. Furthermore, thin superfluid helium\nfilms were the first two-dimensional systems experimentally proven to undergo a\nKosterlitz-Thouless\ntransition to the normal state. \\cite{film} More recently it may\nhave been observed \\cite{Mochel} that on weakly-binding substrates these films\nare the first-known spatially ordered superfluids. \\cite{VAV}\n\nMore precisely, measurements of the third-sound resonance\nfrequency (which is proportional to the square root of the superfluid density)\nof submonolayer helium films on hydrogen and deuterium substrates apparently\nindicate two independent Kosterlitz-Thouless transitions: the usual superfluid\nto normal transition at a temperature $T_{KT}$ that obeys the expected\nuniversal jump relation, \\cite{NK} and a second new transition at a temperature\n$T_c$ which is roughly $0.5\\,\\,T_{KT}$ for all coverages. The second transition\nappears as a sharp (but not discontinuous) rise or dip in the superfluid\ndensity depending on the substrate.\n\nIn an attempt to explain these experimental results we have\nrecently proposed that below the second critical temperature the superfluid\nhelium film is in a spatially ordered phase exhibiting both off-diagonal\n(superfluid) and diagonal (hexatic) long-range order in the one-particle\ndensity matrix. \\cite{PRL} The main idea behind this proposal is that the\nhexatic to fluid transition is known to be a Kosterlitz-Thouless transition\ndriven by disclination unbinding. \\cite{NH}  (Disclinations are defects in the\norientational order of a crystal created by the insertion or removal of a wedge\nof atoms, as shown in Fig.\\ 1.) Therefore, our physical picture of the\nexperiments is that at sufficiently low temperatures the film is in a\nsuperhexatic phase with only a dilute gas of bound vortices and bound\ndisclinations present due to thermal fluctuations. For entropic reasons the\ndisclinations then unbind at $T_c$, leading to a transition from a superhexatic\nto a superfluid phase since the vortices remain bound at this transition and\nthe presence of free disclinations destroys the hexatic long-range order. At\n$T_{KT}$ the vortices then also unbind and the film is finally forced into the\nnormal liquid phase.\n\nOf course, to make sure that the above picture is qualitatively correct we must\nalso consider the interaction between vortices and disclinations. This is even\nmore pressing if one realizes that in a supersolid phase (where all\ndisclination pairs are themselves bound into pairs or triples) this interaction\nis of long range and depends logarithmically on the distance between the two\nkinds of defects. Fortunately, it turns out that this is no longer true in the\nsuperhexatic phase due to the screening of the interaction by the surrounding\ngas of disclination pairs. A renormalization-group analysis actually shows that\nthe vortex-disclination interaction is irrelevant and that the two separate\nKosterlitz-Thouless transitions indeed survive. Nevertheless, the superfluid\ndensity is influenced in a non-universal way by the unbinding of the\ndisclinations and Monte-Carlo simulations even show that on the basis of our\nhypothesis a rough qualitative agreement with the experiments of Chen and\nMochel can be obtained. \\cite{PRL}\n\nHowever, to definitely identify the phase below $T_c$ more detailed information\nis needed. As a first step towards this goal we here present the\ntwo-dimensional hydrodynamic equations of a superhexatic by describing the\nsuperhexatic as a supersolid with free dislocations (i.e.\\ disclination pairs\n\\cite{NH}). As a result of this approach we will also be able to consider the\nhydrodynamics of the supersolid phase, for which there is at present a renewed\ninterest both in the context of Josephson-junction arrays \\cite{Anne} and solid\n$^4$He. \\cite{LG} Moreover, spatially ordered superfluid states have recently\nbeen proposed to be also relevant for the fractional quantum Hall effect,\n\\cite{B} since this effect can be understood as a condensation of composite\nbosons. \\cite{Z} We therefore believe that the methods developed below might,\nif extended to bosons interacting with a Chern-Simons gauge field, also be used\nto obtain a description of the dynamics of such exotic quantum Hall states.\n\nWe have organized the paper in the following manner. In Sec.\\ \\ref{PD} we first\nconsider the normal solid and hexatic phases by formulating a gauge theory that\ndescribes the phonons, the dislocations and the interaction between them. From\nthis theory we then deduce in Sec.\\ \\ref{SOP} for both phases the dynamics of\nthe appropriate hydrodynamic degrees of freedom. In Sec.\\ \\ref{SP} we\nincorporate the effects of the additional superfluid order parameter\n\\cite{fluid} into the hydrodynamic equations derived in Sec.\\ \\ref{SOP} and\ndiscuss the various long-wavelength modes in the supersolid and superhexatic\nphases obtained in this manner. We conclude in Sec.\\ \\ref{DC} with a discussion\non the possible relevance of our work to future experiments on submonolayer\nhelium films and with a physical interpretation of our results.\n\n\\section{GAUGE THEORY OF PHONONS AND DISLOCATIONS}\n\\label{PD}\nIn this section we will derive the long-wavelength (quantum) dynamics of the\nsolid and hexatic phases. The discussion closely follows work by Kleinert,\n\\cite{K1} save that we will not include higher gradient elasticity. This leads\nto a considerable simplification of the theory but implies that we cannot\nproperly treat the dynamics of the disclinations. Fortunately, for our purposes\nonly the dynamics of the dislocations is of importance and this simplification\nis justified.\n\n\\subsection{Solid}\n\\label{S}\nIn the case of an isotropic crystal, the action for the\ndisplacement field $u_i(\\vec{x},\\tau)$ in the presence of a pair of\ndislocations is given by \\cite{K1}\n\\begin{equation}\n\\label{action}\nS[u_i] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n\\left\\{ \\frac{\\rho}{2} (\\partial_{\\tau}u_i - \\beta_i)^2\n           + \\mu \\left(u_{ij} - \\frac{\\beta_{ij} + \\beta_{ji}}{2}\n                 \\right)^2\n           + \\frac{\\lambda}{2} (u_{ii} - \\beta_{ii})^2 \\right\\}~,\n\\end{equation}\nwhere $u_{ij} = (\\partial_i u_j + \\partial_j u_i)\/2$ is the strain tensor,\n$\\mu$ and $\\lambda$ are the usual Lam\\'e coefficients \\cite{La} and $\\rho$ is\nthe average mass density. The unphysical (and singular) contributions arising\nfrom the multivaluedness of $u_i(\\vec{x},\\tau)$ are compensated by the\nquantities $\\beta_j$ and $\\beta_{ij}$ (also known as the `plastic distortion').\nTheir relationship to the defects is best explained by the Volterra\nconstruction. \\cite{K2} Let $\\cal C$ be a small loop bounding a section of two\ndimensional crystal that is excised from the whole (cf. Fig.\\ 2). The edges of\nthe loop are drawn together and form a line $\\cal L$. This line may be time\ndependent, and its definition is not unique. However, the topological defects\n(i.e.\\ two dislocations with opposite Burgers' vectors) associated with the\ndistortion of the surface are always located at the endpoints of $\\cal L$. If\n$\\pm \\vec{B}$ are the Burgers' vectors of the dislocations constituting the\npair and if $\\vec{v}$ is their velocity then\n$\\beta_{ij} = \\delta_i({\\cal L})B_j$ and\n$\\beta_j = - v_i \\delta_i({\\cal L})B_j$. The delta function $\\delta_i({\\cal\nL})$ is singular on the time-dependent Volterra cutting line ${\\cal L}$ of the\ndislocations and is directed along the normal vector. If the cutting line\n${\\cal L}$ is parameterized by $\\vec{x}(s,\\tau)$ with $0 \\leq s \\leq 1$, this\nmeans mathematically that\n\\begin{equation}\n\\delta_i({\\cal L}) = - \\epsilon_{ij} \\int_0^1 ds~\n    \\frac{\\partial x_j(s,\\tau)}{\\partial s}~\n       \\delta(\\vec{x} - \\vec{x}(s,\\tau))~,\n\\end{equation}\nwhere $\\epsilon_{ij}$ is the two-dimensional antisymmetric tensor. Note that\nthe dislocations are assumed to be able to move freely, without any friction,\nthrough the crystal because the equations of motion for the displacement field\nallow for time-dependent solutions that precisely correspond to such evolutions\nof the crystal. \\cite{Na} We will come back to the issue of friction in Sec.\\\n\\ref{SOP} when we consider the effects of dissipation.\n\nWe now first perform a Hubbard-Stratonovich transformation by introducing the\nauxillary variable $\\vec p$ (representing the momentum density) and adding the\nquadratic term\n\\begin{eqnarray}\n\\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n  \\frac{1}{2\\rho} (p_i - i\\rho(\\partial_{\\tau}u_i - \\beta_i))^2\n                                                 \\nonumber\n\\end{eqnarray}\nto the action, which may now be rewritten as\n\\begin{eqnarray}\nS[p_i,u_i] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n\\left\\{ \\frac{p_i^2}{2\\rho} \\right.\n          &+& \\mu \\left(u_{ij} - \\frac{\\beta_{ij} +\n                                       \\beta_{ji}}{2} \\right)^2\n                                                 \\nonumber \\\\\n          &+& \\left. \\frac{\\lambda}{2} (u_{ii} - \\beta_{ii})^2\n           - ip_i(\\partial_{\\tau}u_i - \\beta_i) \\right\\}~.\n\\end{eqnarray}\nIntegrating out $\\vec{p}$ would return the original action up to an unimportant\nconstant. In a similar manner we then also introduce the symmetric stress\ntensor $\\sigma_{ij}$, to decouple the terms quadratic in the strain. This\nresults in\n\\begin{eqnarray}\nS[p_i,\\sigma_{ij},u_i] =\n              \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n\\left\\{ \\frac{p_i^2}{2\\rho} \\right.\n          &+& \\frac{1}{4\\mu} \\left(\\sigma_{ij}^2 -\n                \\frac{\\nu}{1+\\nu} \\sigma_{ii}^2 \\right)\n                                                \\nonumber \\\\\n&-& \\left. ip_i(\\partial_{\\tau}u_i - \\beta_i)\n         + i\\sigma_{ij} \\left( u_{ij} -\n                        \\frac{\\beta_{ij} + \\beta_{ji}}{2}\n                                            \\right) \\right\\}~,\n\\end{eqnarray}\nwith $\\nu = \\lambda\/(2\\mu + \\lambda)$. The partition function is now given by\nthe functional integral\n\\begin{equation}\nZ = \\int d[p_i] \\int d[\\sigma_{ij}] \\int d[u_i]~\n  \\exp \\left\\{-\\frac{1}{\\hbar} S[p_i,\\sigma_{ij},u_i] \\right\\}~,\n\\end{equation}\nwhere the integration over $\\sigma_{ij}$ is only over the symmetrical part\nsince we have not included higher gradient elasticity.\n\nWe can now perform the integration over the displacement field. Because the\naction is linear in $u_i$ this simply leads to the constraint\n\\begin{equation}\n\\label{con}\n\\partial_{\\tau}p_j = \\partial_i \\sigma_{ij}~.\n\\end{equation}\nThis constraint can be automatically satisfied if we introduce the vector field\n$A_j$ and the tensor field $A_{ij}$ by setting\n$\\sigma_{ij} = \\epsilon_{ik} \\partial_k A_j\n             + \\epsilon_{ki} \\partial_\\tau A_{kj}$\nand $p_j = \\epsilon_{ki} \\partial_i A_{kj}$. Substituting these relations into\nthe action we find that the interaction between the gauge fields (i.e.\\ the\nphonons) and the dislocations is\ngiven by\n\\begin{equation}\n\\label{Sint}\nS_{int}[A_{ij},A_j] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n   \\{ -iA_i \\alpha_i + iA_{ij} J_{ij} \\}~,\n\\end{equation}\nwhere after several partial integrations the unphysical singularities of\n$\\beta_i$ and $\\beta_{ij}$ have disappeared and only the dislocation density\nand the dislocation current density\nremain. Introducing also the function\n$\\delta({\\cal P}) = \\delta(\\vec{x}(1,\\tau)) -\n                                 \\delta(\\vec{x}(0,\\tau))$,\nwhich denotes the difference between a delta function at one end of the cutting\nline ${\\cal L}$ and a delta function at the other end, these densities and\ncurrents can conveniently be written as\n$\\alpha_j = \\delta({\\cal P})B_j$ and\n$J_{ij} = - v_i \\delta({\\cal P})B_j$, respectively. As a direct consequence of\nthe above definitions they obey the conservation law\n\\begin{equation}\n\\label{Claw}\n\\partial_{\\tau} \\alpha_j = \\partial_i J_{ij}~.\n\\end{equation}\n\nIn addition, the dynamics of the phonons is determined by the  remaining\nquadratic terms in the action which expressed in terms of the gauge fields\n$A_j$ and $A_{ij}$ yield\n\\begin{equation}\nS_0[A_{ij},A_j] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n  \\left\\{ \\frac{(\\epsilon_{ik} \\partial_k A_{ij})^2}{2\\rho}\n          + \\frac{1}{4\\mu} \\left(\\sigma_{ij}^2 -\n               \\frac{\\nu}{1+\\nu} \\sigma_{ii}^2 \\right) \\right\\}~,\n\\end{equation}\nwith $\\sigma_{ij}$ equal to\n$\\epsilon_{ik}(\\partial_k A_j - \\partial_\\tau  A_{kj})$.\nComparing this result with Eq.\\ (\\ref{action}) we observe that the stress and\nthe physical part of the strain\n$u^{Phys}_{ij} \\equiv u_{ij} - (\\beta_{ij} + \\beta_{ji})\/2$\nare related by\n$\\sigma_{ij} = 2\\mu u^{Phys}_{ij} + \\lambda \\delta_{ij}\n                                                u^{Phys}_{kk}$\nand therefore by\n\\begin{equation}\n\\label{uphys}\nu^{Phys}_{ij} = \\frac{1}{2\\mu} \\left( \\sigma_{ij}\n            - \\frac{\\nu}{1+\\nu} \\delta_{ij} \\sigma_{kk} \\right)~.\n\\end{equation}\nWe will have need of the latter relation in Sec.\\ \\ref{SOP}, when we discuss\nhydrodynamics. A more formal way to justify it is to add to the action\n$S[u_i]$ a source term\n\\begin{eqnarray}\n\\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n  K_{ij} \\left(u_{ij} - \\frac{\\beta_{ij} + \\beta_{ji}}{2} \\right)\n    = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n                              K_{ij} u^{Phys}_{ij}   \\nonumber\n\\end{eqnarray}\nand perform the same manipulations as before. We then find that the source\n$K_{ij}$ indeed couples linearly to the right-hand side of Eq.\\ (\\ref{uphys}).\n\nFollowing Kleinert, we now notice that the above theory has\na gauge symmetry as a result of the fact that the gauge fields\n$A_i$ and $A_{ij}$ are not uniquely determined if the stresses $\\sigma_{ij}$\nand momenta $p_i$ are known. Indeed,\n$\\sigma_{ij}$ and $p_i$ are invariant under the gauge transformation $A_i\n\\rightarrow A_i + \\partial_{\\tau} \\Lambda_i$ and $A_{ij} \\rightarrow A_{ij} +\n\\partial_i \\Lambda_j$. Hence $S_0[A_{ij},A_j]$ is also invariant. Moreover, due\nto the conservation law in Eq.\\ (\\ref{Claw}), the interaction\n$S_{int}[A_{ij},A_j]$ is invariant too.\n\nTo calculate the partition function we therefore need some gauge-fixing\nprocedure. The symmetry of $\\sigma_{ij}$ requires that\n\\begin{equation}\n\\epsilon_{ij} \\sigma_{ij} = \\partial_j A_j\n                               - \\partial_{\\tau}(A_{jj}) = 0~.\n\\end{equation}\nWe would now like to write the gauge fields as the appropriate derivatives of\nunconstrained fields. Using the above gauge symmetry we can always take\n$A_i = \\epsilon_{ij} \\partial_j \\chi$ and $A_{ii}=0$.\nThis, however, does not completely fix the gauge because these conditions are\nstill invariant under the smaller group of transformations\n$\\chi \\rightarrow \\chi + \\partial_{\\tau} \\Lambda$ and\n$A_{ij} \\rightarrow A_{ij} + \\partial_i (\\epsilon_{jk}\n                             \\partial_k \\Lambda)$.\nTo see more clearly the consequences of this residual symmetry we\nexpand $A_{ij}$ into its longitudinal and transverse components (with respect\nto both indices), i.e.\n\\begin{equation}\nA_{ij} = \\partial_i (\\partial_j A^{LL})\n         + \\partial_i (\\epsilon_{jk} \\partial_k A^{LT})\n         + \\epsilon_{ik} \\partial_k (\\partial_j A^{TL})\n         + \\epsilon_{ik} \\partial_k\n                             (\\epsilon_{jl} \\partial_l A^{TT})~,\n\\end{equation}\nwhere we have introduced four new fields. The tracelessness of $A_{ij}$ can\nthen be fulfilled by taking $A^{LL} = -A^{TT}$. In addition, the residual gauge\nsymmetry can now be written as\n$\\chi \\rightarrow \\chi + \\partial_{\\tau} \\Lambda$ and\n$A^{LT} \\rightarrow A^{LT} + \\Lambda$. This shows that instead of the fields\n$\\chi$ and $A^{LT}$ we must use the gauge-invariant field $\\chi' \\equiv \\chi\n- \\partial_{\\tau} A^{LT}$ together with $\\Lambda$ as integration variables. The\nassociated change of\nmeasure can be incorporated in the normalization and the same is true for the\n`volume' $\\int d[\\Lambda]$ of the residual gauge group because the action is\ngauge invariant and therefore cannot depend on $\\Lambda$. After this\ngauge-fixing procedure the partition function thus becomes\n\\begin{equation}\nZ = \\int d[A^{TT}] \\int d[A^{TL}] \\int d[\\chi']~\n  \\exp \\left\\{-\\frac{1}{\\hbar} \\left(S_0[A^{TT},A^{TL},\\chi'] +\n                 S_{int}[A^{TT},A^{TL},\\chi'] \\right) \\right\\}~.\n\\end{equation}\nNote that we are left with three physical degrees of freedom, which is the\ncorrect number in two dimensions since $\\sigma_{ij}$ and $p_i$ contain in\nprinciple a total of five degrees of freedom but we have two constraints in\nEq.\\ (\\ref{con}). Note also that the transformation from $\\sigma_{ij}$ and\n$p_i$ to $A^{TT}$, $A^{TL}$ and $\\chi'$ is a linear one so that the Jacobian\ninvolved in the calculation of the partition function is simply an unimportant\nconstant. In particular, the stress is given by\n\\begin{equation}\n\\label{stress}\n\\sigma_{ij}= \\epsilon_{ik}\\epsilon_{j\\ell}\\partial_k \\partial_\\ell \\chi'\n  + \\partial_\\tau\\left(\n    \\partial_i \\partial_j A^{TL}\n  + \\epsilon_{ik} \\partial_k \\partial_j A^{TT}\n  + \\epsilon_{jk} \\partial_k \\partial_i A^{TT}\n        \\right)~,\n\\end{equation}\nwhich is manifestly symmetric in $i$ and $j$.\n\nA straightforward calculation now shows that the free part of the action is\n\\begin{eqnarray}\n\\label{s0}\nS_0[A^{TT},A^{TL},\\chi'] &=&\n   \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n     \\left\\{ \\frac{1}{2\\mu} (\\partial_{\\tau} \\partial^2 A^{TT})^2\n           + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TT})^2\n     \\right. \\nonumber \\\\\n &+& \\left.\n     \\frac{1}{4\\mu(1+\\nu)} (\\partial_{\\tau} \\partial^2 A^{TL})^2\n           + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TL})^2\n           + \\frac{1}{4\\mu(1+\\nu)} (\\partial^2 \\chi')^2\n     \\right. \\nonumber \\\\\n &-& \\left.\n     \\frac{1}{2\\mu} \\frac{\\nu}{1+\\nu}\n        (\\partial_{\\tau} \\partial^2 A^{TL})(\\partial^2 \\chi')\n     \\right\\}~.\n\\end{eqnarray}\nIt contains four modes: The part involving $A^{TT}$ has a pair of modes\n(corresponding to $\\pm \\vec{k}$) with\n$\\omega^2 = \\mu \\vec{k}^2\/\\rho$. These modes therefore represent the transverse\nphonons with a speed of sound $\\sqrt{\\mu\/\\rho}$. The part involving $A^{TL}$\nand $\\chi'$ has another pair of modes with a dispersion obeying\n$\\omega^2 = (2\\mu + \\lambda) \\vec{k}^2\/\\rho$.\nThese represent the longitudinal phonons with a speed of sound $\\sqrt{(2\\mu +\n\\lambda)\/\\rho}$. Interestingly, these results can be understood much more\neasily if we introduce the field\n\\begin{equation}\n\\label{chidp}\n\\chi'' \\equiv \\chi' - \\nu \\partial_{\\tau} A^{TL}~,\n\\end{equation}\nsince then the above action becomes\n\\begin{eqnarray}\n\\label{uncoupled}\nS_0[A^{TT},A^{TL},\\chi''] &=&\n   \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n     \\left\\{ \\frac{1}{2\\mu} (\\partial_{\\tau} \\partial^2 A^{TT})^2\n           + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TT})^2\n     \\right. \\nonumber \\\\\n &+& \\left.\n     \\frac{1-\\nu}{4\\mu} (\\partial_{\\tau} \\partial^2 A^{TL})^2\n           + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TL})^2\n           + \\frac{1}{4\\mu(1+\\nu)} (\\partial^2 \\chi'')^2\n     \\right\\}~,\n\\end{eqnarray}\nso that the fields are completely uncoupled. Notice that the $\\chi''$ field has\nno kinetic term, which explains why we obtained above only four modes instead\nof six, as might have been expected in first instance.\n\nFurthermore, if we introduce the usual Burgers' field $\\vec{b}(\\vec{x},\\tau)$\nfor the total dislocation density, which is nothing more than the sum of the\ndensity $\\alpha_i$ over all dislocation pairs, then the interaction with the\ndislocations acquires the form\n\\begin{equation}\nS_{int}[A^{TT},A^{TL},\\chi'] =\n  \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    i \\left\\{ \\chi' \\epsilon_{ij} \\partial_j b_i\n              - A^{TT} \\partial_{\\tau} \\partial_i b_i\n      \\right\\}~,\n\\end{equation}\nwhere we have made use of Eq.\\ (\\ref{Claw}) to express the longitudinal part\nof the current density $J_{ij}$ in terms of the time derivative of $b_i$.\nDecomposing $\\vec{b}$ into its transverse and longitudinal parts, i.e.\\\n$b_i = \\partial_i b^L + \\epsilon_{ij} \\partial_j b^T$, the interaction finally\nbecomes\n\\begin{equation}\n\\label{int}\nS_{int}[A^{TT},A^{TL},\\chi'] =\n  \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    i \\left\\{ \\chi' \\partial^2 b^T\n              - A^{TT} \\partial_{\\tau} \\partial^2 b^L\n      \\right\\}~.\n\\end{equation}\nThe total action $S = S_0 + S_{int}$ reduces for time-independent\nconfigurations to the one we previously used for a discussion of the critical\nproperties of the superhexatic. \\cite{PRL} Integrating out the fields $A^{TT}$\nand $\\chi'$ we can now find the time-dependent interaction among the\ndislocations. Physically, these interactions are thus associated with phonon\nexchange and the time dependence arises due to the finite speeds of sound. This\npicture also explains why the effective action for $\\chi'$ contains just\none pair of modes: The self-interaction of the\ntransverse dislocation density can only be mediated by\nlongitudinal phonons.\n\n\\subsection{Hexatic}\n\\label{Hex}\nUp to this point the dislocation density has not been an independent\ndynamical variable, since we have specified the positions of the dislocations\nat all times and thus neglected the influence of the phonon dynamics on\ntheir motion. However, to describe the hexatic phase we want to\nintegrate out the dislocations in the\nplasma (or continuous) approximation. \\cite{NH} For that we need the free\naction of the field $\\vec{b}$. Here we can again make use of the results\nobtained by Kleinert, who showed that the energy associated with the nonlinear\nstresses at the heart of the defect can be lumped into a `core contribution' to\nthe action. \\cite{K1,K3} In our notation this contribution becomes\n\\begin{eqnarray}\n\\label{dislo}\nS_0[b_i] &=& \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{E_c}{2} b_i\n     \\left( \\frac{\\rho}{\\mu}\n            \\frac{\\partial_{\\tau}^2}{\\partial^2} + 1 \\right) b_i\n                                                     \\nonumber \\\\\n &=& \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{E_c}{2} \\left\\{\n       b^T \\left( \\frac{\\rho}{\\mu} \\partial_{\\tau}^2\n                                 + \\partial^2 \\right) b^T\n     + b^L \\left( \\frac{\\rho}{\\mu} \\partial_{\\tau}^2\n                                 + \\partial^2 \\right) b^L\n                   \\right\\} ~.\n\\end{eqnarray}\nThis action represents free propagation of the dislocation density fluctuations\nwhich, as mentioned previously, are permitted by the classical equations of\nmotion \\cite{Na} and neglects dissipative coupling of the dislocation cores to\nthe phonons.\n\nIntegrating out the Burgers' field using Eqs.\\ (\\ref{int}) and (\\ref{dislo}),\nwe obtain the following results. The effective action for $A^{TT}$ becomes\n\\begin{eqnarray}\nS^{eff}[A^{TT}] =\n  \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\left\\{ \\frac{1}{2\\mu} (\\partial_{\\tau} \\partial^2 A^{TT})^2\n    \\right. + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TT})^2\n                                 \\hspace*{1.0in} \\nonumber \\\\\n   + \\left.\n     \\frac{1}{2E_c} (\\partial_{\\tau} \\partial_i A^{TT})\n       \\left(\n         \\frac{\\rho}{\\mu}\\frac{\\partial_{\\tau}^2}{\\partial^2} + 1\n       \\right)^{-1}\n                    (\\partial_{\\tau} \\partial_i A^{TT})\n    \\right\\}~.\n\\end{eqnarray}\nAs shown in Fig.\\ 3a, it contains two pairs of modes which for\n$\\vec{k}^2 \\gg \\mu\/2E_c$ all have a dispersion obeying\n$\\omega^2 \\simeq \\mu \\vec{k}^2\/\\rho$. However, for\n$\\vec{k}^2 \\ll \\mu\/2E_c$, one pair of modes has a dispersion\n$\\omega^2 \\simeq \\mu^2\/E_c\\rho + 2\\mu\\vec{k}^2\/\\rho$ with a gap whereas\nthe other pair of modes is gapless with\n$\\omega^2 \\simeq 2E_c \\vec{k}^4\/\\rho$. This is consistent with our expectation\nthat in the hexatic phase there should only be one pair of transverse\ngapless modes with a softer dispersion than that of the transverse phonon\nmodes in a true solid.\n\nThe effective action for $\\chi'$ and $A^{TL}$ in the hexatic phase is\n\\begin{eqnarray}\nS^{eff}[A^{TL},\\chi'] &=&\n   \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n     \\left\\{\n     \\frac{1}{4\\mu(1+\\nu)} (\\partial_{\\tau} \\partial^2 A^{TL})^2\n     \\right.\n          + \\frac{1}{2\\rho} (\\partial_i \\partial^2 A^{TL})^2\n                                                   \\nonumber \\\\\n         &+& \\frac{1}{4\\mu(1+\\nu)} (\\partial^2 \\chi')^2\n          - \\frac{1}{2\\mu} \\frac{\\nu}{1+\\nu}\n            (\\partial_{\\tau} \\partial^2 A^{TL})(\\partial^2 \\chi')\n                                                   \\nonumber \\\\\n  &+& \\left.\n      \\frac{1}{2E_c} (\\partial_i \\chi')\n       \\left(\n         \\frac{\\rho}{\\mu}\\frac{\\partial_{\\tau}^2}{\\partial^2} + 1\n       \\right)^{-1} (\\partial_i \\chi')\n       \\right\\}~.\n\\end{eqnarray}\nIntegrating out also $A^{TL}$ we finally arrive at the effective action for\n$\\chi'$. It reads\n\\begin{eqnarray}\nS^{eff}[\\chi'] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n     \\left\\{ \\frac{1}{4\\mu(1+\\nu)} (\\partial^2 \\chi')^2\n       + \\frac{1}{2E_c} (\\partial_i \\chi')\n           \\left(\n             \\frac{\\rho}{\\mu}\\frac{\\partial_{\\tau}^2}{\\partial^2}\n                                                              + 1\n           \\right)^{-1} (\\partial_i \\chi')\n     \\right.       \\hspace*{0.5in}        \\nonumber \\\\\n     \\left.\n       + \\frac{1}{2} \\left(\\frac{1}{2\\mu}\n                              \\frac{\\nu}{1+\\nu} \\right)^2\n          (\\partial_{\\tau} \\partial^2 \\chi')\n            \\left( \\frac{\\partial_{\\tau}^2}{2\\mu(1+\\nu)}\n                      + \\frac{\\partial^2}{\\rho} \\right)^{-1}\n             (\\partial_{\\tau} \\partial^2 \\chi')\n     \\right\\}\n\\end{eqnarray}\nand also contains two pairs of modes (cf. Fig.\\ 3b).\nFor $\\vec{k}^2 \\gg \\mu\/2E_c$ we\nrecover of course the ordinary sound dispersions\n$\\omega^2 \\simeq 2\\mu\\vec{k^2}\/(\\rho(1-\\nu))\n                            = (2\\mu +\\lambda)\\vec{k}^2\/\\rho$\nand $\\omega^2 \\simeq \\mu\\vec{k}^2\/\\rho$. However, in the limit\n$\\vec{k}^2 \\ll \\mu\/2E_c$ these evolve into a pair of gapped modes with\n$\\omega^2 \\simeq 2\\mu^2\/(E_c\\rho(1-\\nu))$ and a pair of propagating modes\nwith $\\omega^2 \\simeq 2\\mu(1+\\nu)\\vec{k}^2\/\\rho$, respectively.\nClearly, the same mode structure is\nalso present in the effective action for $A^{TL}$ (obtained by integrating out\n$\\chi'$ instead of $A^{TL}$) which indicates that in the\nhexatic phase the longitudinal velocity is renormalized downwards to\n$\\sqrt{2\\mu(1+\\nu)\/\\rho}$.\n\n\\section{HYDRODYNAMICS OF SPATIALLY ORDERED PHASES}\n\\label{SOP}\nWe now turn to the linear hydrodynamics of the solid and hexatic phases that\nfollows from the theory presented above. For the sake of clarity, and because\nit will turn out to be less important for our purposes, we will not discuss\ntemperature fluctuations in the following. However, having derived the\nrelevant energy\ndensities in Secs. \\ref{S} and \\ref{Hex} it is in principle straightforward to\ninclude temperature fluctuations in our theory and, in particular, to arrive at\nthe extension of the hydrodynamic equations presented below that is required if\none wants to consider also the hydrodynamic mode due to energy conservation.\nAfter the equations of motion for the hydrodynamic variables are determined,\nwe can find the propagating and diffusive modes. This is done as before,\nby Fourier transforming the equations of motion and determining the dispersion\n$\\omega(k)$.\nPropagating modes appear as complex roots of a characteristic equation and\nwill always occur in pairs. Each physically distinct propagating excitation\nsuch as longitudinal or transverse sound corresponds therefore to two\nroots or modes.\n\nWe start by considering the mass-density fluctuation $\\delta\\rho$ above the\naverage mass density $\\rho$ and initially neglect the possible presence of\nvacancies and interstitials. To lowest order in the strain, the density\nfluctuation equals $-\\rho u^{Phys}_{ii}$ so up to that order we obtain\n\\begin{equation}\n\\partial_{\\tau} \\delta\\rho = - \\rho \\partial_{\\tau} u^{Phys}_{ii}\n  = - \\frac{\\rho}{2(\\mu + \\lambda)} \\partial_{\\tau}\n                                                  \\sigma_{ii}~,\n\\end{equation}\nif we make use of Eq.\\ (\\ref{uphys}) to relate the stress and the strain. Using\nalso the decomposition\n$\\sigma_{ii} = \\partial^2 \\chi' + \\partial_{\\tau}\n                                             \\partial^2 A^{TL}$ from\nEq.\\ (\\ref{stress})\nwe may write this as a pair of continuity equations\n\\begin{mathletters}\n\\label{hydro}\n\\begin{equation}\n\\partial_{\\tau} \\delta\\rho = \\partial_i g_i^L~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau} g_j^L = \\partial_i \\pi_{ij}^D~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nwhere the longitudinal momentum density is given by\n\\begin{equation}\ng_i^L = - \\frac{\\rho}{2(\\mu + \\lambda)} \\partial_{\\tau}\n           \\partial_i (\\chi' + \\partial_{\\tau} A^{TL})\n\\end{equation}\nand the diagonal part of the stress tensor by\n\\begin{equation}\n\\pi_{ij}^D = - \\frac{\\rho \\delta_{ij}}{2(\\mu + \\lambda)}\n             \\partial_{\\tau}^2 (\\chi' + \\partial_{\\tau} A^{TL})~.\n\\end{equation}\n\nIn the absence of defects, the hydrodynamic quantities $g_i^L$ and\n$\\pi_{ij}^D$ are precisely equal to the longitudinal part of $p_i$ and the\ndiagonal part of $\\sigma_{ij}$, respectively. This can be seen in the\nfollowing manner. In an ideal solid we have no defects, and variation\nof the action in Eq.\\ (\\ref{uncoupled}) gives $\\chi''=0$ or\n$\\chi'=\\nu \\partial_\\tau A^{TL}$, which we may use to eliminate $\\chi'$.\nThe equation of motion for $A^{TL}$ generated in this way is\n\\begin{equation}\n\\label{ATLmotion}\n\\partial_{\\tau}^2 A^{TL} = - \\frac{2\\mu + \\lambda}{\\rho}\n                                          \\partial^2 A^{TL}.\n\\end{equation}\nIf we substitute this back into the definitions of $g_i^L$ and $\\pi_{ij}^D$\nwe obtain the longitudinal part of $p_i$ and the\ndiagonal part of $\\sigma_{ij}$ as given in section \\ref{PD}.\nIn the presence of defects with their own dynamics this is no longer true,\nsince the dislocation density couples to the gauge fields and alters\nthe equations of motion. To avoid confusion about this point we have,\ntherefore, introduced a new notation for the hydrodynamic\nmomentum density and stress tensor which we will use for the rest of\nthe paper.\n\nWe also note that the above equations\nare not Galilean invariant and are therefore only valid in a specific reference\nframe. This is a result of the fact that the gauge theory of Sec.\\ \\ref{PD} has\nimplicitly used the existence of an ideal lattice with respect to which the\ndisplacements $\\vec{u}(\\vec{x},\\tau)$ are defined. \\cite{K2} Hence, the\nprefered reference frame corresponds to that frame in which this ideal lattice\nis at rest. This is the case for all the hydrodynamic equations presented\nbelow.\n\nIn this ideal solid without interstitials or vacancies the pressure\nfluctuation (following from\n$\\pi_{ij}^D = - \\delta_{ij} \\delta p$) equals\n\\begin{equation}\n\\label{press}\n\\delta p = \\frac{\\rho}{2(\\mu + \\lambda)}\n              (1 + \\nu) \\partial_{\\tau}^3 A^{TL}\n         = \\frac{\\rho}{2\\mu + \\lambda} \\partial_{\\tau}^3\n                                                         A^{TL}\n\\end{equation}\nand the mass-density fluctuation becomes\n\\begin{equation}\n\\delta\\rho = - \\frac{\\rho}{2(\\mu + \\lambda)}\n              (1 + \\nu) \\partial_{\\tau} \\partial^2 A^{TL}\n            = - \\frac{\\rho}{2\\mu + \\lambda}\n                        \\partial_{\\tau} \\partial^2 A^{TL}~.\n\\end{equation}\nSubstituting the equation of motion Eq.\\ (\\ref{ATLmotion})\nfor $A^{TL}$ into Eq.\\ (\\ref{press}), we obtain the desired constitutive\nequation\n\\begin{equation}\n\\delta p = \\frac{2\\mu + \\lambda}{\\rho} \\delta\\rho~.\n\\end{equation}\nTogether with the hydrodynamic equations (\\ref{hydro}) this correctly leads to\nthe sound equation\n\\begin{equation}\n\\label{sound}\n\\partial_{\\tau}^2 \\delta\\rho\n    = - \\frac{2\\mu + \\lambda}{\\rho} \\partial^2 \\delta\\rho\n    = - c_{||}^2 \\partial^2 \\delta\\rho~,\n\\end{equation}\nwith $c_{||}$ the longitudinal sound velocity.\n\nHowever, as stressed first by Martin, Parodi, and Pershan \\cite{M} and again by\nZippelius, Halperin, and Nelson \\cite{ZHN} we are not in general allowed to\nassume that the crystal is ideal, without vacancies or interstitials. We must\ninclude the effects of (long-wavelength) fluctuations in the net defect density\n$n_{\\Delta}$, which is defined as the density\nof vacancies minus the density of interstitials. To do so we can make use of\nthe fact that in a hexagonal system these defects can be regarded as a `bound\nstate' of three dislocations with radial Burgers' vectors pointing\nsymmetrically outward (interstitial) or inward (vacancy). \\cite{N} This is\nillustrated for an interstitial in Fig.\\ 4. As a result the interaction of the\nnet defect density with the gauge fields is given by\n\\begin{equation}\nS_{int}[A^{TT},A^{TL},\\chi'] =\n          \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n      i \\frac{\\gamma_{\\Delta}}{\\mu} n_{\\Delta} \\partial^2\n                                                          \\chi'~,\n\\end{equation}\nwhere $V_0$ denotes the area deficit induced by a defect in an otherwise\nperfect crystal and $\\gamma_\\Delta =\\mu {c_{||}^2} {V_0}\/{2}{c_{\\perp}^2}$.\nWe can verify this result by noting that in the static case (and $n_{\\Delta}\n\\rightarrow in_{\\Delta}$ because of our conventions in the imaginary time\nformalism of Sec.\\ \\ref{PD}) the Euler-Lagrange equation for the Airy stress\nfunction, following from the action in Eq.\\ (\\ref{s0}) together with the above\ninteraction, becomes\n$\\partial^2 \\chi = 2(\\mu + \\lambda) V_0 n_{\\Delta}$\nwhich correctly leads to\n$\\int d\\vec{x}~u^{Phys}_{ii} = V_0 \\int d\\vec{x}~n_{\\Delta}$.\nFurthermore, the free action of the defects becomes\n(cf. Eq.\\ (\\ref{dislo}))\n\\begin{equation}\n\\label{free}\nS_0[n_{\\Delta}] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{E_{\\Delta}}{2} n_{\\Delta}\n     \\left( \\frac{\\rho}{\\mu} \\frac{\\partial_{\\tau}^2}{\\partial^2}\n             + 1 \\right) n_{\\Delta}~,\n\\end{equation}\nwith $E_{\\Delta}$ of order $E_c V_0$.\n\nRedoing our calculations with $n_{\\Delta}$ non-zero, we find that\n$n_{\\Delta}$ displaces the $\\chi'$ field. Therefore $\\chi''$ in Eq.\\\n(\\ref{chidp}) is also non-zero. Moreover, we now obtain instead of Eq.\\\n(\\ref{sound}) the coupled set of equations\n\\begin{mathletters}\n\\label{sounds}\n\\begin{equation}\n\\partial_{\\tau}^2 \\delta\\rho =\n   - c_{||}^2 \\left(\n                1 + \\frac{\\nu \\gamma_{\\Delta}^2}\n                                         {E_{\\Delta} \\mu}\n              \\right) \\partial^2 \\delta\\rho\n   + i \\gamma_{\\Delta}\n              \\left(\n                1 - \\frac{2 \\gamma_{\\Delta}^2}\n                                         {E_{\\Delta} \\mu}\n              \\right) \\partial^2 n_{\\Delta}~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau}^2 n_{\\Delta} =\n   - c_{\\perp}^2 \\left(\n                   1 + \\frac{2 \\gamma_{\\Delta}^2}\n                                         {E_{\\Delta} \\mu}\n                 \\right) \\partial^2 n_{\\Delta}\n   + i \\frac{\\gamma_{\\Delta}\\lambda}\n                   {E_{\\Delta}\\rho^2} \\partial^2 \\delta\\rho~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nfor the longitudinal degrees of freedom. Note that the density fluctuation\n$\\delta\\rho$ receives a contribution from both the lattice vibrations as well\nas\nfrom the net defect density, since\n\\begin{equation}\n\\delta\\rho =\n   - \\frac{1}{c_{||}^2} \\partial_{\\tau} \\partial^2 A^{TL}\n   + \\frac{2i \\gamma_{\\Delta}}{c_{||}^2} n_{\\Delta}~.\n\\end{equation}\nAs a result the longitudinal momentum density has also two contributions\n\\begin{equation}\ng^L_i =\n   - \\frac{1}{c_{||}^2} \\partial_i \\partial_{\\tau}^2 A^{TL}\n   - \\frac{2i \\gamma_{\\Delta}}{c_{||}^2} J^L_i~,\n\\end{equation}\nwhere $\\vec{J}^L$ is the longitudinal part of the net defect current density\nobeying the continuity equation\n$\\partial_{\\tau} n_{\\Delta} = - \\partial_i J^L_i$.\n\nThis almost completes our discussion of the hydrodynamical description (without\ndissipation) of the solid phase. However, we have not yet obtained the\ntransverse modes. From our expressions for the strain tensor $u_{ij}$ one can\neasily show that in the solid phase the transverse part of the displacement\nfield is given by\n\\begin{equation}\n\\label{disp}\nu^T_i = \\frac{1}{\\mu} \\epsilon_{ij}\n                           \\partial_j (\\partial_{\\tau} A^{TT})\n      = \\frac{1}{\\rho c_{\\perp}^2} \\epsilon_{ij}\n                           \\partial_j (\\partial_{\\tau} A^{TT})~,\n\\end{equation}\nwhere $c_{\\perp}$ is the transverse speed of sound. Hence, the transverse\ndynamics of the lattice is solely determined by the transverse phonons and we\nhave the additional hydrodynamic equation\n\\begin{equation}\n\\partial_{\\tau}^2 A^{TT} =\n                    - c_{\\perp}^2 \\partial^2 A^{TT}~,\n\\end{equation}\nwhich is completely uncoupled from the previous ones and in particular does not\ndepend on the net defect density $n_{\\Delta}$. Moreover, if we introduce the\nstandard hexatic order parameter $\\vartheta_6$, which is equal to the local\nbond angle and may therefore be written as\n\\begin{equation}\n\\vartheta_6 \\equiv \\frac{1}{2} \\epsilon_{ij} \\partial_i u_j\n            = - \\frac{1}{2\\rho c_{\\perp}^2}\n                     (\\partial_{\\tau} \\partial^2 A^{TT})~,\n\\end{equation}\nthe equation for $A^{TT}$ is equivalent to\n\\begin{equation}\n\\partial_{\\tau}^2 \\vartheta_6 =\n                    - c_{\\perp}^2 \\partial^2 \\vartheta_6~,\n\\end{equation}\nso that $\\vartheta_6$ can also be used to describe the transverse phonons.\n\n{}From Eq.\\ (\\ref{disp}) we also find that the transverse part of the momentum\ndensity is given by\n\\begin{equation}\ng_i^T = \\frac{1}{c_{\\perp}^2} \\epsilon_{ij}\n                   \\partial_j (\\partial_{\\tau}^2 A^{TT})~.\n\\end{equation}\nTherefore the stress tensor has the nondiagonal contribution\n\\begin{equation}\n\\pi_{ij}^{ND} = - \\frac{1}{c_{\\perp}^2} \\epsilon_{ij}\n                                  (\\partial_{\\tau}^3 A^{TT})\n              = 2\\rho c_{\\perp}^2 \\epsilon_{ij} \\vartheta_6~,\n\\end{equation}\nand both the longitudinal as well as the transverse hydrodynamic equations in\nthe solid phase can be summarized by\n\\begin{mathletters}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t} = -\\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\label{momS}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - c^2 \\nabla \\delta\\rho\n      - \\gamma_{\\Delta} \\nabla n_{\\Delta}\n      + 2\\rho c_{\\perp}^2 \\nabla \\times \\vartheta_6~,\n\\end{equation}\n\\begin{equation}\n\\label{transS}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                   \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial n_{\\Delta}}{\\partial t} =\n      -\\nabla \\cdot \\vec{J}~,\n\\end{equation}\n\\begin{equation}\n\\label{longS}\n\\frac{\\partial \\vec{J}}{\\partial t} =\n      - c_{\\Delta}^2 \\nabla n_{\\Delta}\n      + \\gamma \\nabla \\delta\\rho~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nafter a transformation to real time, which in this case not only means that\n$\\tau \\rightarrow it$ but also\n$\\vec{g} \\rightarrow i\\vec{g}$ and\n$n_{\\Delta} \\rightarrow in_{\\Delta}$. Moreover, note that the constants\n$c$, $\\gamma_{\\Delta}$, $c_{\\Delta}$, and $\\gamma$ should here be\ninterpreted as renormalized quantities which are determined in terms of the\nmicroscopic parameters of our gauge theory by a comparison with\nEq.\\ (\\ref{sounds}).\n\nNow we are ready to discuss dissipation. In principle dissipation has already\nbeen included because there is a coupling between the net defect density\n$n_{\\Delta}$ and the phonons. Hence if for example an interstitial were, in a\ndiscrete picture, to tunnel from one location to another there would be a\n`shake up' of the phonon field. However, if we treat $n_{\\Delta}$ as a smooth\ncontinuously varying field, the action in Eq.\\ (\\ref{free}) is quadratic and\nthe bilinear coupling\n$n_{\\Delta} \\partial^2 \\chi'$ produces only mixing of the collective modes\nbut no real dissipation. Therefore, we choose to include effective dissipation\nin the same manner as explained in detail by Zippelius, Halperin and Nelson.\n\\cite{ZHN} Using their notation we first of all add to the right-hand side of\nEq.\\ (\\ref{momS}) the terms\n$(\\eta \\nabla^2 \\vec{g}\n   + \\zeta \\nabla (\\nabla \\cdot \\vec{g}) )\/\\rho$\nassociated with the dissipative part of the stress tensor $\\pi_{ij}$ and\nrepresenting viscous diffusion of the momentum density.\n\nNext the question arises how we need to modify\nEq.\\ (\\ref{longS}). This equation is a result of the fact that we have allowed\nthe dislocations, and hence the interstitials and vacancies, to move freely\nthrough the lattice and used Eq.\\ (\\ref{free}) for the free action of the\ndefects. If the defects effectively experience friction (for example due to\ntheir interaction with the phonons), then it is more appropriate to add a\nLeggett friction term \\cite{TL} and use\n\\begin{equation}\n\\label{diss}\nS_0[n_{\\Delta}] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{E_{\\Delta}}{2} n_{\\Delta}\n     \\left( \\frac{\\rho}{\\mu}\n       \\frac{\\partial_{\\tau}^2}{\\partial^2}\n         + i \\frac{\\rho}{\\mu} \\xi \\partial_{\\tau} + 1\n     \\right) n_{\\Delta}\n\\end{equation}\ninstead. The dispersions then indeed obey\n$\\omega^{\\pm} \\simeq \\pm c_{\\perp} k - i \\xi k^2\/2$ at long wavelengths, and we\nmust add the term\n$\\xi \\nabla (\\nabla \\cdot \\vec{J})$ to the right-hand side of Eq.\\\n(\\ref{longS}). If we further assume that the transverse part of the defect\ncurrent density behaves as in a gas and simply diffuses to zero with a\ndiffusion constant $\\kappa$, we obtain in total\n\\begin{mathletters}\n\\label{HS}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t} = -\\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - \\frac{B}{\\rho} \\nabla \\delta\\rho\n      - \\gamma_{\\Delta} \\nabla n_{\\Delta}\n      + 2\\rho c_{\\perp}^2 \\nabla \\times \\vartheta_6\n      + \\frac{\\eta}{\\rho} \\nabla^2 \\vec{g}\n      + \\frac{\\zeta}{\\rho} \\nabla (\\nabla \\cdot \\vec{g})~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                  \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial n_{\\Delta}}{\\partial t} =\n      -\\nabla \\cdot \\vec{J}~,\n\\end{equation}\n\\begin{equation}\n\\label{diff}\n\\frac{\\partial \\vec{J}}{\\partial t} =\n      - c_{\\Delta}^2 \\nabla n_{\\Delta}\n      + \\gamma \\nabla \\delta\\rho\n      + \\kappa \\nabla^2 \\vec{J}\n      + \\xi \\nabla (\\nabla \\cdot \\vec{J})~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nwith $B = \\rho \\partial p\/\\partial\\rho|_{n_{\\Delta},T} =\n                                          \\rho c^2$\nthe appropriate isothermal bulk modulus in view of the fact that the pressure\nis a function of both the particle density as well as the net defect density.\n{}From thermodynamics we therefore also conclude that\n$\\gamma_{\\Delta} = \\partial p\/\\partial n_{\\Delta}|_{\\rho,T}$.\n\nIt is interesting to point out that these hydrodynamic equations differ from\nthe results obtained by Zippelius, Halperin, and Nelson. In particular, their\nEq.\\ (3.32) differs from our\nEq.\\ (\\ref{diff}) and reads\n\\begin{equation}\n\\vec{J} =\n   - \\Gamma_{\\Delta} \\nabla\n       \\left( \\frac{n_{\\Delta}}{\\chi_{\\Delta}} -\n                                  \\gamma_{\\Delta} \\delta\\rho\n       \\right)~.\n\\end{equation}\nThe difference can easily be traced back to the fact that Zippelius, Halperin,\nand Nelson assume on phenomenological grounds that the dynamics of the net\ndefect density is purely diffusive. Indeed, we exactly reproduce their results\nif we use in our calculation a free action of the form\n\\begin{equation}\nS_0[n_{\\Delta}] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{E_{\\Delta}}{2} n_{\\Delta}\n          \\left(\n            \\frac{i\\partial_{\\tau}}{D_{\\Delta} \\partial^2} + 1\n          \\right) n_{\\Delta}\n\\end{equation}\ninstead of Eq.\\ (\\ref{free}). We can therefore consider the hydrodynamic\nequations of Zippelius, Halperin and Nelson as the overdamped (or classical)\nlimit of our Eq.\\ (\\ref{HS}). Clearly, Kleinert's more microscopic approach\ndoes not lead to purely diffusive but in first instance to propagating behavior\nof\nthe defects, which is appropriate for the quantum crystals of interest in Sec.\\\n\\ref{SP}. We now turn to the modification of the above results in the hexatic\nphase.\n\nIn the hexatic phase there are free dislocations present and $\\chi''$ replaces\n$n_{\\Delta}$ as the appropriate dynamical degree of freedom. To see most\nclearly how this comes about we will work perturbatively in $1\/E_c$. Up to\nfirst order in $1\/E_c$\nthe effective action for $\\chi''$ is\n\\begin{eqnarray}\n\\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n     \\left\\{ \\frac{1}{4\\mu(1+\\nu)} (\\partial^2 \\chi'')^2\n       + \\frac{1}{2E_c} (\\partial_i \\chi'')\n           \\left(\n             \\frac{\\rho}{\\mu}\n             \\frac{\\partial_{\\tau}^2}{\\partial^2} + 1\n           \\right)^{-1} (\\partial_i \\chi'')\n          \\right\\}~,                             \\nonumber\n\\end{eqnarray}\nwhich upon Fourier transformation displays two modes with\n$\\omega^2 = c_{\\perp}^2 \\vec{k}^2 + 2\\mu^2(1+\\nu)\/(E_c\\rho)$. So in the\nlimit $E_c \\rightarrow \\infty$ (which physically means that we are looking at\nthe nonhydrodynamic regime\n$\\vec{k}^2 \\gg \\mu\/2E_c$) we approximately have\n\\begin{equation}\n\\partial_{\\tau}^2 \\chi'' = - c_{\\perp}^2 \\partial^2 \\chi''~,\n\\end{equation}\nwhereas the equations of motion for $A^{TL}$ and $A^{TT}$ are\n\\begin{equation}\n\\partial_{\\tau}^2 A^{TL} =\n                    - c_{||}^2 \\nabla^2 A^{TL}\n                    + \\frac{\\nu}{1-\\nu^2} \\partial_{\\tau} \\chi''\n\\end{equation}\nand\n\\begin{equation}\n\\partial_{\\tau}^2 A^{TT} =\n                    - c_{\\perp}^2 \\nabla^2 A^{TT}~,\n\\end{equation}\nrespectively. For the mass-density fluctuation we now find\n\\begin{equation}\n\\delta\\rho = - \\frac{1}{c_{||}^2}\n            \\partial_{\\tau} \\partial^2 A^{TL}\n          - \\frac{\\rho}{2(\\mu + \\lambda)} \\partial^2 \\chi''\n\\end{equation}\nand for the stress tensor\n\\begin{equation}\n\\pi_{ij}^D = - \\delta_{ij}\n   \\left\\{\n      \\frac{1}{c_{||}^2} \\partial_{\\tau}^3 A^{TL}\n    + \\frac{\\rho}{2(\\mu + \\lambda)} \\partial_{\\tau}^2 \\chi''\n   \\right\\}\n         = - \\delta_{ij}\n   \\left\\{ c_{||}^2 \\delta\\rho\n           + \\frac{1}{2(1+\\nu)} \\partial^2 \\chi''\n   \\right\\}~.\n\\end{equation}\n\nPutting all this together we obtain in first instance the following set of\nhydrodynamic equations for the hexatic phase\n\\begin{mathletters}\n\\begin{equation}\n\\partial_{\\tau} \\delta\\rho = \\nabla \\cdot \\vec{g}^L~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau} \\vec{g}^L = - \\nabla \\delta p\n    = - c_{||}^2 \\nabla \\delta\\rho\n      - \\frac{1}{2(1+\\nu)} \\nabla (\\nabla^2 \\chi'')~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau}^2 \\chi'' = - c_{\\perp}^2 \\nabla^2 \\chi''~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau}^2 A^{TT} =\n                    - c_{\\perp}^2 \\nabla^2 A^{TT}~.\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nCombining the longitudinal and transverse parts as before, this equals\n\\begin{mathletters}\n\\begin{equation}\n\\partial_{\\tau} \\delta\\rho = \\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\label{mom}\n\\partial_{\\tau} \\vec{g} =\n      - c_{||}^2 \\nabla \\delta\\rho\n      - \\frac{1}{2(1+\\nu)} \\nabla (\\nabla^2 \\chi'')\n      + 2\\rho c_{\\perp}^2 \\nabla \\times \\vartheta_6~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau}^2 \\chi'' = - c_{\\perp}^2 \\nabla^2 \\chi''~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau} \\vartheta_6 =\n                   - \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nand clearly reduces to the hydrodynamic equations for the ideal crystal if we\nput $\\chi''=0$.\n\nWe now have to consider how the above picture changes for a finite value of\n$E_c$. Here we can use the results of\nSec.\\ \\ref{Hex}. In the hydrodynamic regime\n$\\vec{k}^2 \\ll \\mu\/2E_c$ we saw that the transverse speed of sound was\nrenormalized to zero, because we found the quadratic (particle-like) dispersion\n$\\omega^2 = 2E_c \\vec{k}^4\/\\rho$. As a result we have for the transverse\npart of the hydrodynamic equations\n\\begin{equation}\n\\partial_{\\tau}^2 A^{TT} =\n                    \\frac{2E_c}{\\rho} \\partial^4 A^{TT}~,\n\\end{equation}\nwhich implies that in the right-hand side of Eq.\\ (\\ref{mom}) we must replace\n$2\\rho c_{\\perp}^2 \\nabla \\times \\vartheta_6$ by\n$- 4 E_c \\vec{e}_z \\times \\nabla (\\nabla^2 \\vartheta_6)$. This gives\n\\begin{mathletters}\n\\begin{equation}\n\\partial_{\\tau} \\vec{g}^T =\n      - 4 E_c \\vec{e}_z \\times \\nabla (\\nabla^2 \\vartheta_6)~,\n\\end{equation}\n\\begin{equation}\n\\partial_{\\tau} \\vartheta_6 =\n                   - \\frac{1}{2\\rho} \\nabla \\times \\vec{g}^T~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nwhich is in complete agreement with Zippelius, Halperin, and Nelson if we\nidentify the Frank constant $K_A$ with $8E_c$.\n\nFor the longitudinal part we need to analyze the dynamics of $\\chi''$ and\n$A^{TL}$. A straightforward calculation shows that the effective action for\nthese fields contains precisely the four modes already found in Sec.\\\n\\ref{Hex}. The propagating modes with $\\omega^2 = 2\\mu(1+\\nu)\n\\vec{k}^2\/\\rho$ obey\n$\\chi' = \\chi'' + \\nu \\partial_{\\tau} A^{TL} =0$ and are therefore indeed\nassociated with density fluctuations proportional to $\\partial^2 \\chi''$. We\nthus need to use a renormalized longitudinal speed of sound equal to\n\\begin{equation}\nc = \\sqrt{\\frac{2\\mu (1+\\nu)}{\\rho}}\n       = \\sqrt{\\frac{2\\mu}{\\rho}\n                  \\frac{2\\mu + 2\\lambda}{2\\mu + \\lambda}}\n\\end{equation}\nthat is always smaller than the longitudinal speed of sound in the solid phase.\nIn fact, this actually exhausts the longitudinal hydrodynamic modes since the\nother modes in the effective action for $\\chi''$ and $A^{TL}$ are gapped. As a\nresult we now obtain in real time the following set of hydrodynamic equations\nfor the hexatic phase\n\\begin{mathletters}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t}\n                                = - \\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\label{momH}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - c^2 \\nabla \\delta\\rho\n      - \\frac{K_A}{2}\n           \\,\\,\\vec{e}_z \\times \\nabla (\\nabla^2 \\vartheta_6)~,\n\\end{equation}\n\\begin{equation}\n\\label{transH}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                    \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nnot including dissipation.\n\nTo include dissipation we again follow Zippelius, Halperin and Nelson and add\nto the right-hand side of Eq.\\ (\\ref{momH}) the terms $(\\eta \\nabla^2 \\vec{g}\n        + \\zeta \\nabla (\\nabla \\cdot \\vec{g}) )\/\\rho$.\nHowever, we do not add the term $\\kappa \\nabla^2 \\vartheta_6$ to the right-hand\nside of Eq.\\ (\\ref{transH}) because, just as in the solid phase, the\ndissipation of the transverse modes is already accounted for in the term $\\eta\n\\nabla^2 \\vec{g}$ that is added to the momentum equation. Put differently, a\nterm of the form $\\kappa \\nabla^2 \\vartheta_6$ can be absorbed by an\nappropriate redefinition of $K_A$ and $\\eta$. Again introducing the isothermal\nbulk modulus\n$B = \\rho dp\/d\\rho|_T = \\rho c^2$ we then find\n\\begin{mathletters}\n\\label{HH}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t}\n                                = - \\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - \\frac{B}{\\rho} \\nabla \\delta\\rho\n      - \\frac{K_A}{2}\n           \\vec{e}_z \\times \\nabla (\\nabla^2 \\vartheta_6)\n      + \\frac{\\eta}{\\rho} \\nabla^2 \\vec{g}\n      + \\frac{\\zeta}{\\rho} \\nabla (\\nabla \\cdot \\vec{g}) ~,\n\\end{equation}\n\\begin{equation}\n\\label{theta}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                    \\frac{1}{2\\rho} \\nabla \\times \\vec{g}\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nas our final result for the hexatic phase. Apart from the absence of a\ndissipative term in Eq.\\ (\\ref{theta}) it agrees with the findings of\nZippelius, Halperin and Nelson and therefore contains the same mode structure\nas derived in that paper. For completeness sake, we mention however that the\nequations of motion for the hexatic order parameter $\\vartheta_6$ can be\nderived from an effective action\n\\begin{equation}\nS^{eff}[\\vartheta_6] = \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n    \\frac{1}{2} \\vartheta_6\n     \\left( 4\\rho \\frac{\\partial_{\\tau}^2}{\\partial^2}\n       + 4i \\eta \\partial_{\\tau} - K_A \\partial^2\n     \\right) \\vartheta_6~,\n\\end{equation}\nthat can easily be understood physically: The first term on the right-hand side\ncorresponds to the kinetic energy\n$\\int d\\vec{x} \\rho (\\partial_{\\tau} \\vec{u})^2\/2$\nof the displacement field. The second term is a Leggett friction term and the\nlast term corresponds to the usual Frank energy, which is responsible for the\nfact that the hexatic to liquid transition is of the Kosterlitz-Thouless type.\n\n\\section{HYDRODYNAMICS OF SUPERFLUID PHASES}\n\\label{SP}\nHaving arrived at the hydrodynamic equations for the solid and hexatic phases,\nour next objective is to incorporate the effects of the additional hydrodynamic\ndegree of freedom associated with the phase of the superfluid order parameter.\nFortunately, from the microscopic theories developed for superfluid liquids\n\\cite{HM} and gases \\cite{KD} it is well known how we should proceed to obtain\nthe hydrodynamic (two-fluid) equations for the superfluid phases starting from\nthe equations for the normal phase. The procedure consists in principle of four\nsteps. First, the total (average) density $\\rho$ of the system is split up into\na normal density $\\rho_n$ and a superfluid density $\\rho_s$. In general these\ndensities are tensors of second rank, but for systems with hexagonal symmetry\nwhich are of interest here they are proportional to the identity $\\delta_{ij}$\nand can be considered as scalars. Second, the total momentum density $\\vec{g}$\nis similarly split up into a normal component\n$\\rho_n \\vec{v}_n$ and a superfluid component $\\rho_s \\vec{v}_s$ with a\nsuperfluid velocity that is purely longitudinal\n($\\nabla \\times \\vec{v}_s = 0$). Third, for an effectively isotropic system the\ndissipative terms in the momentum equation must be generalized to\n\\begin{eqnarray}\n\\eta \\nabla^2 \\vec{v}_n\n      + \\zeta_1 \\frac{\\rho_s}{\\rho}\n                \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n      + \\zeta_2 \\nabla (\\nabla \\cdot \\vec{v}_n)~.    \\nonumber\n\\end{eqnarray}\nFinally, we must add the dynamics of the superfluid velocity, which is\nbasically determined from the Josephson relation and reads\n\\begin{equation}\n\\frac{\\partial \\vec{v}_s}{\\partial t} =\n    - \\frac{B}{\\rho^2} \\nabla \\delta\\rho\n    + \\zeta_3 \\frac{\\rho_s}{\\rho}\n              \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n    + \\zeta_4 \\nabla (\\nabla \\cdot \\vec{v}_n)~,\n\\end{equation}\nwhere $B=\\rho^2 d\\mu\/d\\rho|_T$ is again the isothermal bulk modulus and\n$\\mu$ is the chemical potential per unit mass. We again leave out temperature\nfluctuations since we are primarily interested in third-sound modes, for which\nthese fluctuations are (at least qualitatively) unimportant.\n\n\\subsection{Supersolid}\n\\label{HSS}\nTo apply the above procedure to Eq.\\ (\\ref{HS}) we must realize that we are\nhere in fact already dealing with a two-fluid hydrodynamics. We must therefore\nnot only split up the total momentum density $\\vec{g}$ into a normal and a\nsuperfluid component but also the net defect current, i.e.\\\n$\\vec{J} = \\vec{J}_n + \\vec{J}_s$. Moreover, we have to account for the fact\nthat the chemical potential, just like the pressure, is a function of the\nparticle density and the net defect density. In this manner we arrive at the\nfollowing hydrodynamic equations\n\\begin{mathletters}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t} = -\\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{eqnarray}\n\\label{momrho}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - \\frac{B}{\\rho} \\nabla \\delta\\rho\n     &-& \\gamma_{\\Delta} \\nabla n_{\\Delta}\n      + 2\\rho c_{\\perp}^2 \\nabla \\times \\vartheta_6\n                                                   \\nonumber \\\\\n     &+& \\eta \\nabla^2 \\vec{v}_n\n      + \\zeta_1 \\frac{\\rho_s}{\\rho}\n                \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n      + \\zeta_2 \\nabla (\\nabla \\cdot \\vec{v}_n)~,\n\\end{eqnarray}\n\\begin{equation}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                  \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vec{v}_s}{\\partial t} =\n    - \\frac{B}{\\rho^2} \\nabla \\delta\\rho\n    + \\beta_{\\Delta} \\nabla n_{\\Delta}\n    + \\zeta_3 \\frac{\\rho_s}{\\rho}\n              \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n    + \\zeta_4 \\nabla (\\nabla \\cdot \\vec{v}_n)~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial n_{\\Delta}}{\\partial t} =\n                  -\\nabla \\cdot \\vec{J}~,\n\\end{equation}\n\\begin{equation}\n\\label{momdel}\n\\frac{\\partial \\vec{J}}{\\partial t} =\n      - c_{\\Delta}^2 \\nabla n_{\\Delta}\n      + \\gamma \\nabla \\delta\\rho\n      + \\kappa \\nabla^2 \\vec{J}_n\n      + \\xi_1 \\nabla (\\nabla \\cdot \\vec{J}_s)\n      + \\xi_2 \\nabla (\\nabla \\cdot \\vec{J}_n)~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vec{J}_s}{\\partial t} =\n    - \\frac{B_{\\Delta} \\rho_s}{\\rho^2} \\nabla n_{\\Delta}\n    + \\beta \\rho_s \\nabla \\delta\\rho\n    + \\xi_3 \\frac{\\rho_s}{\\rho}\n            \\nabla (\\nabla \\cdot \\vec{J}_s)\n    + \\xi_4 \\nabla (\\nabla \\cdot \\vec{J}_n)~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nwith\n$\\beta_{\\Delta} = - \\partial \\mu\/\\partial\n                           n_{\\Delta}|_{\\rho,T}$.\nThese represent nine equations for the nine unknown functions $\\delta\\rho$,\n$\\vec{v}_n$, $\\vec{v}_s$, $\\vartheta_6$, $n_{\\Delta}$, $\\vec{J}_n$ and\n$\\vec{J}_s$.\n\nAlthough a complete analysis of the various hydrodynamic modes is now possible,\nwe will consider here only the situation which is most relevant to experiments,\nnamely that the normal part of the two-dimensional system is\nclamped to an underlying substrate. As a result we have\n$\\vec{v}_n = \\vec{J}_n = \\vec{0}$ and\nEqs.\\ (\\ref{momrho}) and (\\ref{momdel}) determining the normal properties of\nthe supersolid are no longer valid. The hydrodynamic equations therefore reduce\nto\n\\begin{mathletters}\n\\label{clamped}\n\\begin{equation}\n\\frac{\\partial^2 \\delta\\rho}{\\partial t^2} =\n    \\frac{B \\rho_s}{\\rho^2} \\nabla^2 \\delta\\rho\n    - \\beta_{\\Delta} \\rho_s \\nabla^2 n_{\\Delta}\n    + \\zeta_3 \\frac{\\rho_s}{\\rho}\n        \\frac{\\partial}{\\partial t}(\\nabla^2 \\delta\\rho)~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial^2 n_{\\Delta}}{\\partial t^2} =\n    \\frac{B_{\\Delta} \\rho_s}{\\rho^2} \\nabla^2 n_{\\Delta}\n    - \\beta \\rho_s \\nabla^2 \\delta\\rho\n    + \\xi_3 \\frac{\\rho_s}{\\rho}\n        \\frac{\\partial}{\\partial t} (\\nabla^2 n_{\\Delta})~.\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nThey contain two pairs of\npropagating modes, which in the limit of a small coupling\nconstant\n$\\beta \\ll BB_{\\Delta}\/\\beta_{\\Delta} \\rho^4$ essentially correspond to\na pair of\nthird-sound modes with $\\delta\\rho$ unequal to zero but a constant net defect\ndensity and a pair of modes with an oscillating net defect density.\n\nOne might have expected that the coupling of a superfluid\ndensity to a propagating defect density  would have resulted\nin one pair of gapped excitations and one pair of gapless excitations\ninstead.  Consider, for example, two identical superfluid layers.\nIf the layers are uncoupled the dynamics of the\nphases $\\vartheta_1$ and $\\vartheta_2$ of the layers is determined by the\naction\n\\begin{equation}\nS_{layers}[\\vartheta_1,\\vartheta_2] =\n \\int_0^{\\hbar \\beta} d\\tau \\int d\\vec{x}~\n   \\left\\{ \\frac{\\rho^2}{2B} (\\partial_{\\tau} \\vartheta_1)^2 +\n           \\frac{\\rho_s}{2} (\\nabla \\vartheta_1)^2\n         + \\frac{\\rho^2}{2B} (\\partial_{\\tau} \\vartheta_2)^2 +\n           \\frac{\\rho_s}{2} (\\nabla \\vartheta_2)^2\n   \\right\\}~,\n\\end{equation}\nwhich clearly has two pairs of\ngapless (third-sound) modes, one pair for each superfluid.\nIf we couple the order parameters by allowing the particles\nto tunnel with an amplitude $-J\/\\rho$\nfrom one layer to the other we must add\na Josephson coupling\n\\begin{equation}\nS_{tunnel}[\\vartheta_1,\\vartheta_2] = - \\int d\\tau \\int d\\vec{x}~\n    J \\cos(\\vartheta_1 - \\vartheta_2)\n\\end{equation}\nto this action.  The hydrodynamics modes couple to form two in-phase and\ntwo out-of-phase excitations.  The\nmodes with $\\vartheta_1$ and $\\vartheta_2$ oscillating out of phase get\ngapped (i.e.\\\n$\\omega^2 \\simeq BJ\/\\rho^2$ for $\\vec{k}^2 \\ll J\/\\rho_s$) and only the\nmodes with $\\vartheta_1$ and $\\vartheta_2$ oscillating in phase remain gapless.\nYet in Eq.\\ (\\ref{clamped}) we find only gapless modes.\n\nThis paradox can be resolved by noting that\nwe have made the standard assumption\\cite{ZHN} that both the\ntotal number of particles and the net number of defects is conserved. Hence,\nafter an atom has tunneled from a lattice site to the position of a vacancy, a\nnew vacancy is created near the original site of the atom.\nThe analogous process for the\ntwo coupled superfluid layers in not simply tunneling of\nindividual atoms from one layer to another, but rather the\nexchange of a pair of atoms in different layers, returning the\nsystem to its original state. Such a process is not a\nJosephson coupling and therefore the modes remain gapless.\nThe existence of separate conservation laws for the particle and defect\ndensity  thus allows in principle two separate broken symmetries.\n\nWe also note in passing\nthat the third-sound modes in Eq.\\ (\\ref{clamped}) are not present\nin the hydrodynamic equations proposed by Andreev and Lifshitz \\cite{A} and\nconsidered in more detail by Liu. \\cite{L} This is a result of the fact that\nthese authors use a somewhat different physical picture for the supersolid\nphase: They assume that the superfluid current density is carried by (Bose\ncondensed) defects and that the normal current density is solely due to lattice\nvibrations. Hence if we take\n$\\vec{v}_n = \\vec{0}$, which in their context means that $\\partial\n\\vec{u}\/\\partial t = \\vec{0}$, only transport of defects is possible and only\nthe latter two modes survive. However, as a consequence of their picture the\nhydrodynamic equations in the (normal) solid phase describe only longitudinal\nand transverse sound modes in an ideal lattice and do not include the effect of\nvacancies or interstitials. As explained above this is incorrect in principle\nand one should at least also allow for a normal current density due to the\nmotion of defects. In addition, we have seen in Sec.\\ \\ref{SOP} that even in\nthe presence of defects the density fluctuations are equal to $-\\rho\nu^{Phys}_{ii}$. It is therefore perfectly reasonable that if there is\nsuperfluid mass transport possible in the solid, it can be caused both by the\nmotion of defects and by lattice vibrations. Indeed, as an existence proof of\nthis latter possibility we can for instance consider superfluid $^4$He in a\nweak periodic and commensurate potential, which is clearly a supersolid without\ndefects.\n\nWhile it is generically possible to have both density and\ndefect superfluid modes, we might expect however, for realistic films on\nrealistic substrates, that in a supersolid it may be harder for particles to\nperform ring exchanges\\cite{K2,ring} than for vacancies\nto exchange positions. Thus, {\\it a priori}, we might expect the effective\nsuperfluid stiffness for the density fluctuations to be smaller than that\nof the vacancies, perhaps to the point where the former is entirely absent.\n\n\\subsection{Superhexatic}\nWe next turn to the superhexatic phase. In a similar manner as in Sec.\\\n\\ref{HSS} we obtain from Eq.\\ (\\ref{HH}) the full set of hydrodynamic equations\n\\begin{mathletters}\n\\begin{equation}\n\\frac{\\partial \\delta\\rho}{\\partial t}\n                                = - \\nabla \\cdot \\vec{g}~,\n\\end{equation}\n\\begin{eqnarray}\n\\frac{\\partial \\vec{g}}{\\partial t} =\n      - \\frac{B}{\\rho} \\nabla \\delta\\rho\n     &-& \\frac{K_A}{2}\n           \\vec{e}_z \\times \\nabla (\\nabla^2 \\vartheta_6)\n                                                  \\nonumber \\\\\n     &+& \\eta \\nabla^2 \\vec{v}_n\n      + \\zeta_1 \\frac{\\rho_s}{\\rho}\n                \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n      + \\zeta_2 \\nabla (\\nabla \\cdot \\vec{v}_n)~,\n\\end{eqnarray}\n\\begin{equation}\n\\label{theta6}\n\\frac{\\partial \\vartheta_6}{\\partial t} =\n                    \\frac{1}{2\\rho} \\nabla \\times \\vec{g}~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial \\vec{v}_s}{\\partial t} =\n    - \\frac{B}{\\rho^2} \\nabla \\delta\\rho\n    + \\zeta_3 \\frac{\\rho_s}{\\rho}\n              \\nabla (\\nabla \\cdot (\\vec{v}_s - \\vec{v}_n))\n    + \\zeta_4 \\nabla (\\nabla \\cdot \\vec{v}_n)~,\n\\end{equation}\n\\end{mathletters}\n\n\\noindent\nthat leads to the usual two-fluid hydrodynamics of a superfluid if we omit Eq.\\\n(\\ref{theta6}) and put $\\vartheta_6 = 0$. Therefore these equations allow for\nfirst and second sound, \\cite{second} and for a pair of transverse modes\ninvolving $\\vec{v}_n^T$ and $\\vartheta_6$ which are either dispersive or\npropagating depending on the sign of\n$\\Delta = K_A\/4\\rho - (\\eta\/\\rho_n)^2$: If $\\Delta \\leq 0$ we have two\npurely dispersive modes with\n$\\omega^{\\pm} = -i(\\eta\/\\rho_n \\pm \\sqrt{-\\Delta}) \\vec{k}^2\/2$,\nwheras if $\\Delta > 0$ we have two propagating modes and the particle-like\ndispersion\n$\\omega^{\\pm} = \\pm \\sqrt{\\Delta}~ \\vec{k}^2\/2\n                            - i (\\eta\/\\rho_n) \\vec{k}^2\/2$.\nHowever, considering again the case $\\vec{v}_n = \\vec{0}$ the hydrodynamic\nequations now simply reduce to\n\\begin{equation}\n\\frac{\\partial^2 \\delta\\rho}{\\partial t^2} =\n    \\frac{B \\rho_s}{\\rho^2} \\nabla^2 \\delta\\rho\n    + \\zeta_3 \\frac{\\rho_s}{\\rho}\n        \\frac{\\partial}{\\partial t}(\\nabla^2 \\delta\\rho)~,\n\\end{equation}\nwhich contains only a pair of third-sound modes with the velocity\n$c_3 = \\sqrt{B\\rho_s\/\\rho^2}$ and the diffusion constant\n$D_3 = \\zeta_3 \\rho_s\/\\rho$.\n\n\\section{CONCLUSIONS AND DISCUSSION}\n\\label{DC}\nIn this paper we have derived the hydrodynamic equations for the  supersolid\nand superhexatic phases of a neutral two-dimensional Bose fluid. For the\nsupersolid these equations are rather complex, since they incorporate the\neffects of defect motion and lattice vibrations on both the normal and\nsuperfluid parts of the momentum density. Our physical picture for the\ninfluence on the superfluid part is roughly speaking that in a mean-field\ntheory the condensate wavefunction $\\Psi(\\vec{x},t)$ obeys the Schr\\\"odinger\nequation\n\\begin{equation}\ni\\hbar \\frac{\\partial \\Psi(\\vec{x},t)}{\\partial t} =\n   \\left\\{\n     - \\frac{\\hbar^2 \\nabla^2}{2m}\n     + \\int d\\vec{x}'~ V(\\vec{x}-\\vec{x}') n(\\vec{x}',t)\n   \\right\\} \\Psi(\\vec{x},t)~,\n\\end{equation}\nwhere $m$ is the mass of the Bose particles and $V(\\vec{x}-\\vec{x}')$ is their\ninteraction. In addition, $n(\\vec{x},t)$ is the particle density which will be\ndetermined by an additional mean-field theory that, for a supersolid, shows the\ninstability associated with the formation of a density wave. Hence the\n(thermal) average\n$\\langle n(\\vec{x},t) \\rangle$ is periodic in space and independent of time. As\na result the condensate wavefunction is, if we neglect density fluctuations,\nalso periodic and we have indeed both diagonal as well as off-diagonal\nlong-range order. Fluctuations in the density, however, induce variations in\nthe phase of the wavefunction and therefore in the superfluid velocity. Because\nthese density fluctuations can be caused by both lattice vibrations and\noscillations in the net defect density we conclude that both mechanisms can\nlead to superfluid motion. Together with the existence of a conservation law\nfor the net number of defects, this explains from a more microscopic view\nwhy we found two third-sound modes and two modes with an oscillatory net defect\ndensity in the case of a supersolid adsorbed onto a substrate.\n\nFor the superhexatic phase we have shown that the hexatic long-range order\nleads to an additional (as compared to the superfluid) hydrodynamic degree of\nfreedom that affects only the transverse modes and is therefore at long\nwavelengths decoupled from the superfluid momentum density. This can also be\nunderstood from the above picture, since variations in the orientational order\nparameter $\\vartheta_6$ do not lead to density fluctuations in first instance.\nAs a result we find on a substrate only two third-sound modes and thus at the\nhydrodynamic level of description nothing to distinguish the superhexatic from\nthe superfluid. Although this is in agreement with the experiments of Chen and\nMochel, who indeed only observe one third-sound branch below the second\ncritical temperature $T_c$, it is unfortunate for the purpose of suggesting a\npossible identification of the superhexatic phase. On the basis of our results\nwe can, however, conclude that a more microscopic probe is needed if one wants\nto detect the orientational order present in a superhexatic helium film. In our\nopinion this appears to be an important, but also difficult experimental\nchallenge.\n\nFinally, we would also like to point out the possible relevance of our results\nto the recent experiments with bulk solid $^4$He. \\cite{LG} In these\nexperiments Lengua and Goodkind observe at sufficiently high frequencies an\nadditional (resonant) attenuation and velocity change of sound. Moreover, they\nnotice that their data can be explained by a simple model of two coupled wave\nequations which turns out to be identical to the longitudonal part of our\nsolid hydrodynamics derived in\nSec.\\ \\ref{SOP}. Because our two-dimensional hydrodynamics should be able to\ndescribe the propagation of sound perpendicular to the c-axis of hcp $^4$He,\nthis confirms the conjecture of Lengua and Goodkind that the collective mode\nobserved is associated with the motion of defects. For a more detailed\ndiscussion of the coupling between sound and the defects one should of course\nconsider the fully three-dimensional situation and include the anisotropy of\nthe hcp crystal. Work in this direction is in progress.\n\n\\section*{ACKNOWLEDGMENTS}\nThis research was supported by Grant No.\\ DMR-9502555 and DMR-9416906\nfrom the National\nScience Foundation, the ESF Network on Quantum Fluids and Solids,\nthe Swedish Natural Science Research Council and the\nStichting voor Fundamenteel Onderzoek der Materie (FOM) which is\nfinancially supported by the Nederlandse Organisatie voor\nWetenschappelijk Onderzoek (NWO).\nWe thank Huug van Beelen, Henk van Beijeren, Michel Bijlsma,\nReyer Jochemsen, and Anne van Otterlo for stimulating and helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe (Borel-Moore) homology of a smooth manifold possesses a canonical ring structure in which the product is represented by intersection of cycle classes.\nThe Chow groups of a smooth algebraic variety over the complex numbers also admit a ring structure, and the intersection products are compatible via the cycle class map \\cite[Cor.~19.2]{Ful}.  In both contexts, a well-behaved intersection theory for cycles fails to extend to spaces with singularities.\n\nIn topology, this motivated the definition of cohomology and its cup product operation, which in some sense isolates a subset of the space of cycles (with a different equivalence relation) which can be intersected even if the ambient space has singularities.  There are several analogues in algebraic geometry.\nThe Friedlander-Lawson theory of algebraic cocycles \\cite{FLcocycle} is a geometric approach built from finite correspondences to projective spaces, and similar constructions underlie the motivic cohomology of Friedlander-Voevodsky \\cite {FV}.  \nThe operational Chow cohomology of Fulton \\cite{Ful}, on the other hand, adopts a more formal approach. \n\n\nThe intersection homology groups of Goresky and MacPherson \\cite{GM1} provide an\ninterpolation between cohomology and homology: at least for a normal space $X$ there is a sequence of groups:\n$$H^{\\dim X -*}(X) = IH^{\\overline 0}_* (X) \\to \\cdots \\to IH^{\\overline p}_*(X) \\to \\cdots \\to IH^{\\overline t}_*(X) = H_*(X)$$\nfactoring the cap product map.  The decoration $\\overline p$, called the perversity, is a sequence of integers which prescribes how cycles may meet the strata in a suitable stratification of the (possibly singular) space $X$; in the display above it increases from left to right.  Each intersection homology group $IH^{\\overline p}_r(X)$ arises as the homology of a complex of chains (either simplicial chains with respect to a triangulation \\cite{GM1}, or singular chains \\cite{King}) of perversity $\\overline p$.\n\nOne of the most interesting features of this theory is the existence of intersection pairings\n$$IH^{\\overline p}_r(X) \\otimes IH^{\\overline q}_s(X) \\to IH^{\\overline p + \\overline q}_{r+s - \\dim(X)}(X)$$\n(provided $\\overline p + \\overline q \\leq \\overline t$) generalizing the cap product pairing between cohomology and homology, and providing a generalization of Poincar\\'e duality to singular spaces.\n\nThe author and Eric Friedlander \\cite{intsing} have defined an algebraic cycle counterpart to the geometric approach of Goresky-MacPherson.  In particular we have defined perverse Borel-Moore motivic homology groups $H^{\\overline p}_{m}(X, \\mathbb{Z}(r))$ (for a stratified variety $X$ and a perversity $\\overline p$) with a cycle class map to the Goresky-MacPherson theory in Chow degree: $H^{\\overline p}_{2r}(X, \\mathbb{Z}(r)) \\to IH^{\\overline p}_{2r}(X, \\mathbb{Z})$ \\cite[Defn.~2.3, Prop.~2.5]{intsing}.  We have constructed also a perverse variation of motivic cohomology $H^{i,s, \\overline p}(X)$ and pairings \\cite[Defn.~5.3, Prop.~6.13]{intsing}:\n$$\\cap : H^{i,s, \\overline p}(X) \\otimes H^{\\overline q}_m (X, \\mathbb{Z}(r) ) \\to H^{\\overline p + \\overline q}_{m-i}(X, \\mathbb{Z}(r-s)) .$$\nThere is a canonical morphism $H^{i, s, \\overline p}(X) \\to H^{\\overline p}_{2 d -i}(X, \\mathbb{Z}(d -s))$ \\cite[Cor.~6.14]{intsing} with $d = \\dim (X)$.  To define an intersection product\n$$H^{\\overline p}_{m}(X, \\mathbb{Z}(r)) \\otimes H^{\\overline q}_{n}(X, \\mathbb{Z}(s)) \\to H^{\\overline p + \\overline q}_{m+n - 2 d}(X, \\mathbb{Z}(r+s - d))$$\n(i.e., extend $\\cap$), one needs to involve the stratification in a more substantial way.\n\nIn this paper we approach the problem of defining such a product in Chow degree ($m=2r, n = 2s$) by considering the following (somewhat vague) question: if a singular variety $X$ can be resolved by a smooth variety $\\widetilde X$, to what extent does the intersection product on the Chow groups of $\\widetilde X$ provide a sensible intersection theory for algebraic cycles on $X$?  As one can already see from the case of a proper birational morphism $\\pi : \\widetilde X \\to X$ between smooth varieties, the push-forward of the intersection formed on $\\widetilde X$ is in general different from the intersection formed on $X$.\nOn the other hand, one also sees that for a blowup $\\widetilde X \\to X$ of a smooth variety $X$ along a smooth subvariety, the push-forward of the intersection formed on $\\widetilde X$ agrees with the intersection formed on $X$ if the cycles have controlled incidence with $Y$; see Proposition \\ref{smooth case} for a precise statement.\n\nOur main results establish cases in which intersections formed on a resolution provide well-defined products on modifications of Chow groups: instead of considering cycles modulo rational equivalence, certain strata are singled out by the resolution, and both the cycles and the equivalences among them are required to have controlled incidence with the strata.  If $k$ is a field of characteristic zero, then any $k$-variety $X$ may be resolved by a sequence of blowups along smooth centers $\\pi : \\widetilde X \\to X$.  From such a resolution we define a stratification of $X$ and construct an intersection pairing on perverse Chow groups (i.e., a map $A_{r, \\overline p} (X) \\otimes A_{s, \\overline q}(X) \\to A_{r+s-d}(X)$, possibly with $\\mathbb{Q}$ coefficients) in the following settings:\n\\begin{itemize}\n\\item $s=\\dim(X)-1$, i.e., one of the cycles is a divisor (Theorem \\ref{divisor pairing});\n\\item $r=s=2$ and $\\dim(X)=4$, i.e., there is an intersection pairing on 2-cycles on a fourfold (Theorem \\ref{4fold}); and\n\\item $\\dim(X_{sing})=1$ and $r,s$ arbitrary (Theorem \\ref{one dim sing}).\n\\end{itemize}\nThe pairing is obtained by pushing forward the intersection product of the proper transforms, i.e., is given by $\\alpha, \\beta \\mapsto \\pi_* (\\widetilde \\alpha \\cdot \\widetilde \\beta)$.  The stratification is expressed in terms of the geometry of the resolution, and our arguments are accordingly geometric in nature.  In all three cases, the basic idea is to understand how (certain) rational equivalences behave under proper transform.  For concreteness we explain the idea for 2-cycles $\\alpha, \\beta$ of complementary perversities $\\overline{p}, \\overline{q}$ on a fourfold $X$.  If one has $\\alpha \\sim_{\\overline p} \\alpha'$ on $X$, then $e_\\alpha := \\widetilde \\alpha - \\widetilde {\\alpha'}$ is a cycle on $\\widetilde X$ supported over the singular locus of $X$, with the support controlled by the perversity function $\\overline p$.\nThe perversity $\\overline{p}$ provides enough control over the error term $e_\\alpha$ that we can find a cycle which is both rationally equivalent to some multiple of $e_\\alpha$ and, using the complementarity of the perversity condition $\\overline q$, disjoint from $\\widetilde{\\beta}$ (or $e_\\beta$).\nThen, at least with rational coefficients, we have the vanishings  $e_\\alpha \\cdot \\widetilde \\beta = \\widetilde \\alpha \\cdot e_\\beta = e_\\alpha \\cdot e_\\beta = 0$ in $A_0(\\widetilde X)$, so that $\\pi_* (\\widetilde \\alpha \\cdot \\widetilde \\beta )$ and $\\pi_*(\\widetilde \\alpha' \\cdot \\widetilde \\beta')$ coincide as classes in $A_0(X)_\\mathbb{Q}$.\nFor divisors the arguments are more conceptual and less technical, but the main point is to move error terms arising from (certain) rational equivalences away from cycles of complementary perversity.  When the singular locus of $X$ is one-dimensional, we also employ intersection theory on the exceptional components and subvarieties therein.\n\n\nFor certain pairs of cycles, our product agrees with the one defined by Goresky-MacPherson via the cycle class map, but we do not know if this holds in general.  Another basic question is the dependence on the resolution.  See \\S \\ref{sec:further} for further discussion of these questions.\n\nIn \\S \\ref{sec:cycles} we establish background and notation.  In particular, we discuss various procedures for constructing stratifications from resolutions; later we point out which constructions are necessary in the different contexts.  The next three sections are devoted to the three situations mentioned above: in \\S \\ref{sec:divisors} we show the resolution provides a sensible intersection theory for divisors, in \\S \\ref{sec:onedimsing} we handle the case $\\dim(X_{sing}) =1$, and in \\S \\ref{sec:4fold} we study 2-cycles on a fourfold.  In \\S \\ref{need integral fibers} we show, through an explicit example, that an ``obvious\" coarsening of our stratification, namely by the fiber dimension of the resolution, is insufficiently fine for the resolution to provide a decent intersection product.\n\n\\medskip\n\\textbf{Conventions.} Throughout we work with schemes separated and of finite type over a field $k$.  A variety is an integral $k$-scheme.\n\n\\medskip\n\\textbf{Acknowledgments.} \nThe idea that resolutions might provide pairings on perverse cycle groups emerged in conversations with Eric Friedlander.\nThe author wishes to thank him for his encouragement and interest in this project.  The author was partially supported by National Science Foundation Award DMS-0966589.\n\n\n\\section{Cycles, equivalence relations, and stratifications from resolutions} \\label{sec:cycles}\n\nIn this section we adapt some of the basic features of Goresky-MacPherson intersection homology to our algebraic context.  This reproduces some material from \\cite[\\S 2]{intsing}.  Then we discuss stratifications obtained from resolutions.  Finally we demonstrate that our proposal for intersections on a singular variety recovers the usual intersection theory for the blowup of a smooth variety along a smooth subvariety.\n\nA \\textit{stratified variety} is a variety $X$ (say of dimension $d$) equipped with a filtration by closed subsets $X^d \\hookrightarrow X^{d-1} \\hookrightarrow \\cdots \\hookrightarrow X^2 \\hookrightarrow X^1 \\hookrightarrow X$ such that $X^i$ has codimension at least $i$ in $X$.  This is usually called a filtered space, which determines a stratified space; since we do not deal with more general stratifications, we ignore the difference.\nA \\textit{perversity} is a non-decreasing sequence of integers $p_1, p_2, \\ldots, p_d$ such that $p_1=0$ and, for all $i$, $p_{i+1}$ equals either $p_i$ or $p_i+1$.  Perversities are denoted $\\overline{p}, \\overline{q}$, etc.  The perversities we consider range from the zero perversity $\\overline{0}$ with $p_i=0$ for all $i$, to the top perversity $\\overline{t}$, with $p_i=i-1$ for all $i$.\n\nLet $Z_r(X)$ denote the group of $r$-dimensional algebraic cycles on $X$.  If $X$ is stratified and $\\overline p$ is a perversity, we say an $r$-cycle $\\alpha$ is \\textit{of perversity $\\overline{p}$} (or satisfies the perversity condition $\\overline{p}$) if for all $i$, the dimension of the intersection $|\\alpha| \\cap X^i$ is no larger than $r-i +p_i$.  If the codimension of $X^i$ in $X$ is exactly $i$, then the perversity of a cycle measures its failure to meet properly the closed sets occurring in the stratification of $X$.  Let $Z_{r, \\overline{p}}(X) \\subset Z_r(X)$ denote the group of $r$-dimensional cycles of perversity $\\overline{p}$ on the stratified variety $X$.  Typically $X^1$ is the singular locus of $X$, and then the condition $p_1=0$ means that no component of the cycle is contained in the singular locus.\n\nIn \\cite[Prop.~2.4]{intsing} we defined and characterized a notion of $\\overline p$-rational equivalence of algebraic $r$-cycles: we identify two elements of $Z_{r, \\overline p}(X)$ if they can be connected by an $\\mathbb{A}^1$-family of $r$-cycles of perversity $\\overline p$.  Here we introduce a modification.\nWe say the cycles $\\alpha, \\alpha' \\in Z_{r, \\overline{p}}(X)$ are \\textit{weakly rationally equivalent as perversity $\\overline{p}$ cycles} (or are weakly $\\overline{p}$-rationally equivalent), written $\\alpha \\sim_{w, \\overline{p}} \\alpha'$, if there is an equation of rational equivalence all of whose terms satisfy the perversity condition $\\overline{p}$.  Explicitly this means there exist subvarieties $W_1, \\ldots, W_a$ of dimension $r+1$ in $X$, and rational functions $g_i : W_i \\dashrightarrow \\bb{P}^1$, such that $\\alpha - \\alpha' = \\sum_i [g_i(0)] - [g_i(\\infty)]$ in $Z_{r, \\overline{p}}(X)$ (i.e., for all $i$, both $[g_i(0)]$ and $[g_i(\\infty)]$ are in $Z_{r, \\overline{p}}(X)$).  \n\nIn contrast to the definition of $A_{r, \\overline p}(X)$, here we do \\textit{not} require that for every $t \\in \\mathbb{A}^1 \\subset \\bb{P}^1$, the cycle associated to the fiber $[g_i(t)]$ satisfies the perversity condition.  We allow the $g_i$'s to have ``bad\" fibers, but we require that $\\alpha$ and $\\alpha'$ are related by ``good\" fibers.  We could equivalently require that the condition of \\cite[Prop.~2.4(3)]{intsing} is satisfied with $\\mathbb{A}^1$ replaced by an open subset (containing at least two $k$-points) therein.\nWe let $A^w_{r, \\overline{p}}(X)$ denote the group of $r$-cycles of perversity $\\overline{p}$ modulo weak $\\overline{p}$-rational equivalence.  \nThe main reason for introducing another equivalence relation is that our arguments ``naturally\" construct pairings on the groups $A^w_{r, \\overline p}(X)$.\nSince there is a canonical morphism $A_{r, \\overline{p}}(X) \\to A^w_{r, \\overline p}(X)$, we immediately obtain pairings on the groups $A_{r, \\overline p}(X)$.  Note also the map $A_{r, \\overline p}(X) \\to IH^{\\overline p}_{2r}(X, \\mathbb{Z})$ factors through $A^w_{r, \\overline p}(X)$: the proof of \\cite[Prop.~2.5]{intsing} only requires $\\mathcal W \\hookrightarrow X \\times \\bb{P}^1$ to have good fibers over $0, \\infty \\in \\bb{P}^1$, not all $t \\in \\bb{P}^1$.\n\n\n\\medskip\n\n\\textbf{Resolutions.} Let $X$ be a variety over a field $k$ of characteristic zero.  The celebrated result of Hironaka asserts the singularities of $X$ can be resolved by a sequence of blowups along smooth centers which are normally flat in their ambient spaces, i.e., the resolution can be expressed as a composition $\\pi: \\widetilde{X} = X_f \\to \\cdots \\to X_{n+1} \\to X_n \\to \\cdots \\to X = X_0$ where (for all $n = 0, \\ldots, f-1$) $X_{n+1} = \\Bl_{C_n} X_n$, $C_n \\hookrightarrow X_n$ is smooth, and $X_n$ is normally flat along $C_n$ \\cite{Hironaka}.  We will call a resolution admitting such an expression a \\textit{strong resolution.}\n\n\\begin{definition} The exceptional locus $\\widetilde{E} \\hookrightarrow \\widetilde{X}$ of the resolution is simply the preimage of the singular locus of $X$.  We say a subvariety $V \\hookrightarrow \\widetilde{X}$ is \\textit{exceptional} if it is contained in the exceptional locus of the resolution.  We say an exceptional subvariety $V \\hookrightarrow \\widetilde{X}$ \\textit{first appears} on $X_{n+1} = \\Bl_{C_n}X_n$ if there is a subvariety $\\overline{V} \\hookrightarrow X_{n+1}$ such that the morphism $V \\hookrightarrow \\widetilde{X} \\to X_{n+1}$ factors through a birational morphism $V \\to \\overline{V}$, and there is no such subvariety of $X_n$.  The concept of first appearance depends on the sequence of centers used to construct the resolution. \\end{definition}\n\n\\begin{notation} If $C_n \\hookrightarrow X_n$ is a center occurring in the resolution, let $\\widetilde{E}_{C_n} \\subseteq \\widetilde{E}$ denote the union of those exceptional divisors $E$ for which the canonical morphism $E \\to X$ factors through $C_n$, and let $\\widetilde{E}_{\\text{dom }C_n} \\subseteq \\widetilde{E}_{C_n}$ denote the union of those divisors for which the canonical morphism $E \\to C_n$ is dominant.   \\end{notation}\n\n\\begin{definition}[stratification via resolution] \\label{strat res defn} \nWe describe several ways a center $C_n$ may contribute to a stratification of the variety being resolved.\nFirst we consider what happens ``below\" a center $C_n$.  \n\\begin{BC}\n\\item If $C_n \\to X$ has generic dimension $g$ over its $m$-dimensional image $W \\hookrightarrow X$, then $W \\hookrightarrow X^{d-m}$; and for each $e \\geq 1$ the locus $W' \\hookrightarrow W$ over which $C_n \\to W$ has fiber dimension $\\geq g+e$ is placed in $X^{d-m+e+1}$.  \n\\item If $g \\geq 1$, the generic fiber of $C_n \\to W$ is integral; the locus over which $C_n \\to W$ fails to have an integral fiber of dimension $g$ is placed in $X^{d-m+1}$.\n\\item The singularities of $W$ are placed in $X^{d-m+1}$.\n\\end{BC}\nNext we consider what happens ``above\" a center.\n\\begin{AC}\n\\item For a general point $c \\in C_n$, the number of irreducible components of ${(\\widetilde{E}_{\\text{dom }C_n})}_c$ is equal to the number of components of $\\widetilde{E}_{\\text{dom }C_n}$; where the former exceeds the latter is a closed set denoted $R_n \\hookrightarrow C_n$.  \nIf the image $V$ of $R_n$ in $W$ is not dense, we place this image in $X^{d-m+e}$, where $e$ is the codimension of $V$ in $W$ (hence $d-m+e$ is the codimension of $V$ in $X$). \n\\end{AC}\nLet $\\pi : \\widetilde X \\to X$ be a resolution expressed as a sequence of blowups along $C_n \\hookrightarrow X_n$.  For any of the four constructions listed above, we say a stratification of $X$ \\textit{satisfies condition (C) with respect to $\\pi$} if it refines the stratification obtained by applying construction (C) to all of the centers $C_n$ occurring in the resolution.  \\end{definition}\n\n\\begin{remark} Let $\\pi : \\widetilde X \\to X$ be a resolution.  If a stratification satisfies (BC-1) with respect to $\\pi$, then it refines the ``fiber dimension\" stratification given by specifying $X^i$ is the closed subset along which the fibers of $\\pi$ have dimension at least $i-1$.\n\nA stratification satisfying (BC-1), (BC-2), (BC-3), and (AC) may not have the property that $X^i \\setminus X^{i+1}$ is smooth since we do not, for example, necessarily place the incidences of the components of $X_{sing}$ in a smaller stratum.  For an explicit example in which the incidences of the components of the singular locus are not detected by the conditions above, see the projective example of Section \\ref{need integral fibers}.\n\nIt would be very interesting to characterize intrinsically the strata which arise from the conditions described above.\n\\end{remark}\n\n\n\\medskip\n\\textbf{Error terms.} Suppose $\\alpha \\sim_{w, \\overline{p}} \\alpha'$ and \\begin{equation} \\label{alpha equiv} \\alpha - \\alpha' = \\sum_{f_i \\in k(S_i)} [f_i(0)] - [f_i(\\infty)] \\end{equation}\nis an equation of rational equivalence in $Z_{r, \\overline{p}}(X)$.  Viewing the $f_i$'s as rational functions on the proper transforms of the $S_i$'s, we obtain an equation of rational equivalence $\\widetilde{\\alpha} + e_\\alpha - \\widetilde{\\alpha'} - e_{\\alpha'} = \\sum_{f_i \\in k(\\widetilde{S_i})} [f_i(0)] - [f_i(\\infty)]$ in $Z_r(\\widetilde{X})$, where $e_\\alpha$ is exceptional for some $\\widetilde{S_i} \\to S_i$ and is supported over the image of the exceptional locus of $\\widetilde{z} \\to z$ for $z$ some term in the equation $\\ref{alpha equiv}$ (and similarly for $e_{\\alpha'}$).  In this situation we refer to $e_\\alpha$ and $e_{\\alpha'}$ as the ``error terms.\"\nWe write $e(\\alpha, \\alpha')$ for the union of the supports of the error terms, and typically $e_\\alpha$ will denote a component of the support of $e(\\alpha, \\alpha')$.\n\n\\medskip\n\\textbf{Normality.} There is no loss of generality in assuming $X$ is normal.  For if $X$ is not necessarily normal and $\\nu : X^\\nu \\to X$ is the normalization, and $\\pi : \\widetilde{X^\\nu} \\to X^\\nu$ is the result of applying a resolution algorithm to $X^\\nu$, we simply declare the stratification of $X$ to be the image via $\\nu$ of the stratification of $X^\\nu$ (induced by $\\pi$), augmented by declaring $X^1$ is the singular locus of $X$.  Note that $\\nu \\circ \\pi$ is probably not the result of applying a resolution algorithm to $X$ itself.\n\nFor $\\alpha \\in Z_{r, \\overline{p}}(X)$, let $\\alpha^\\nu \\in Z_r(X^\\nu)$ denote its proper transform.  By definition, $\\alpha \\in Z_{r, \\overline{p}}(X)$ implies $\\alpha^\\nu \\in Z_{r, \\overline{p}}(X^\\nu)$.  If $\\alpha \\in Z_{r, \\overline{p}}(X)$, then $\\dim ( \\nu^{-1}(\\alpha \\cap X^1)) = \\dim (\\alpha \\cap X^1) = r-1$ since $\\nu$ is finite and $\\alpha$ is not contained in $X^1$ (since $p_1=0$).  Therefore $\\overline{p}$-rational equivalence on $X$ implies $\\overline{p}$-rational equivalence on $X^\\nu$.  Let $\\nu^* : A_{r, \\overline{p}}(X) \\to A_{r, \\overline{p}}(X^\\nu)$ denote the morphism induced by proper transform.\n\nNow suppose we can define a pairing $A_{r, \\overline{p}}(X^\\nu) \\times A_{s, \\overline{q}}(X^\\nu) \\to A_{r+s-d}(X^\\nu)$ when $\\overline{p} + \\overline{q} = \\overline{t}$.  Then the composition\n$$A_{r, \\overline{p}}(X) \\times A_{s, \\overline{q}}(X) \\xrightarrow{\\nu^* \\times \\nu^*} A_{r, \\overline{p}}(X^\\nu) \\times A_{s, \\overline{q}}(X^\\nu) \\to A_{r+s-d}(X^\\nu) \\xrightarrow{\\nu_*} A_{r+s-d}(X)$$\ndefines the pairing for $X$.\n\nMoreover, the condition $p_1=0$ implies the generic point of any error term factors through $X^2$: we have $\\dim (\\alpha \\cap X^1) \\leq r-1$, and $\\pi$ is finite over $X \\setminus X^2$, hence $\\dim (\\pi^{-1} (\\alpha \\cap X^1 \\setminus X^2)) \\leq r-1$ and $\\pi^{-1}(\\alpha \\cap X^1 \\setminus X^2)$ cannot support any error terms.\n\n\nIf $X$ is not normal, the stratification ``prescribed by the resolution\" is the one prescribed by $\\widetilde{X^\\nu} \\to X^\\nu \\to X$.  We assume $X$ is normal and hence $X^2$ consists of the singular locus of $X$.\n\n\\medskip\nThe following proposition demonstrates our proposal is sensible when the ``resolution\" is the blowup of a smooth variety along a smooth subvariety.  The method of proof also shows the condition $\\overline p + \\overline q \\leq \\overline t$ cannot in general be weakened.\n\n\\begin{proposition} \\label{smooth case}\nLet $Y \\hookrightarrow X$ be a closed immersion of smooth varieties of codimension $c$.  Let $\\alpha, \\beta \\hookrightarrow X$ be cycles of dimensions $r,s$ satisfying\n$$\\dim(\\alpha \\cap Y) \\leq r-c+p \\ , \\ \\dim(\\beta \\cap Y) \\leq s-c+q,$$\nwith $p+q \\leq c-1$.  Let $\\pi: \\widetilde X \\to X$ be the blowup of $X$ along $Y$.  Then $\\alpha \\cdot \\beta = \\pi_*( \\widetilde \\alpha \\cdot \\widetilde \\beta)$ as cycle classes on\n$|\\alpha| \\cap |\\beta|$ and hence on $X$.\n\\end{proposition}\n\n\\begin{proof} We have a cartesian diagram:\n\n$$\\xymatrix{ E \\ar[r]^-j \\ar[d]_-g & \\widetilde X\\ar[d]^-\\pi \\\\ Y \\ar[r] & X .\\\\}$$\nLet $z \\in A_*(E)$ denote the class of the canonical $g$-ample line bundle.  Then $A_*(E)$ is $A_*(Y) [z]$ modulo a relation involving the Chern classes of the normal bundle $N_YX$; we will need that $g^*$ is a ring homomorphism, and that $g_* (z^n) = 0$ for $n \\leq c-2$.\n\n We set $\\pi^* \\alpha = \\widetilde \\alpha + e_\\alpha$ and $\\pi^* \\beta = \\widetilde \\beta + e_\\beta$ in $A_*(\\widetilde X)$; precise formulas for $e_\\alpha, e_\\beta$ are available \\cite[Thm.~6.7]{Ful}, but we will just need that $\\pi_* e_\\alpha = \\pi_*e_\\beta=0$.  Since $\\alpha \\cdot \\beta = (\\pi_* \\pi^* \\alpha) \\cdot \\beta = \\pi_* (\\pi^* \\alpha \\cdot \\pi^* \\beta)$, it suffices to show the terms $\\widetilde \\alpha \\cdot e_\\beta, e_\\alpha \\cdot \\widetilde \\beta,$ and $e_\\alpha \\cdot e_\\beta$ vanish after $\\pi_*$ is applied.\n\nNow we analyze the term $\\widetilde \\alpha \\cdot e_\\beta$; the treatment of the term $e_\\alpha \\cdot \\widetilde \\beta$ merely involves interchanging the roles of $\\alpha$ and $\\beta$.  We utilize the Chow rings of $Y$ and $E$ and show the vanishing of $g_* ( j^* (\\widetilde \\alpha) \\cdot e_\\beta )$, which implies $\\pi_* (\\widetilde \\alpha \\cdot e_\\beta) = 0$.  Since $j^* (\\widetilde \\alpha)$ is an $(r-1)$-dimensional cycle supported over the ($\\leq (r-c+p)$-dimensional) set $\\alpha \\cap Y$, it can be expressed as\n$$j^* (\\widetilde \\alpha) = z^p g^*( a_{r-c+p} ) + z^{p-1}  g^* (a_{r-c+p-1} ) + \\cdots +  g^*( a_{r-c} )$$\nwhere $a_i$ is a class in $A_i(Y)$.  Similarly, since $e_\\beta$ is an $s$-dimensional cycle supported over $\\beta \\cap Y$, we have an expression\n$$e_\\beta = z^{q-1} g^* (b_{s-c+q}) + z^{q-2} g^* (b_{s-c+q-1}) + \\cdots + g^* ( b_{s-c+1})$$\nwith $b_i \\in A_i(Y)$.  Therefore $ j^* (\\widetilde \\alpha) \\cdot e_\\beta$ is a sum of terms of the shape $z^n g^* ( c_n )$ (with $c_n \\in A_{r+s-d+1+n-c}(Y)$) with $n \\leq p+q -1$.  Now the projection formula for $g$, the formula $g_* (z^n) = 0$ for $n \\leq c-2$, and the hypothesis $p+q \\leq c-1$ together imply that $g_* ( j^* (\\widetilde \\alpha) \\cdot e_\\beta) =0 $, as desired.\n\nTo show the vanishing of the term $e_\\alpha \\cdot e_\\beta$, we note that\n$$e_\\alpha \\cdot_{\\widetilde X} e_\\beta =  j_* ( c_1(N_{E} \\widetilde X) \\cap (e_\\alpha \\cdot_E e_\\beta) ),$$\nwhere we have used a subscript to indicate on which variety we calculate the product.  The largest $z$-degree in $e_\\alpha \\cdot_E  e_\\beta$ is $p+q-2$, and $ c_1(N_{E} \\widetilde X) = -z$, so the same argument as in the previous paragraph shows the vanishing of $g_* ( c_1(N_{E} \\widetilde X) \\cap (e_\\alpha \\cdot e_\\beta) )$, hence $\\pi_* (e_\\alpha \\cdot_{\\widetilde X} e_\\beta) = 0$.\n\\end{proof}\n\n\n\\begin{remark}\nThere does not seem to be a simple inductive argument which allows one to conclude Proposition \\ref{smooth case} holds for a composition of blowups along smooth centers $\\pi: X_2 = \\text{Bl}_{Y_1}(X_1) \\xrightarrow{\\pi_2} X_1 = \\text{Bl}_{Y_0}(X_0) \\xrightarrow{\\pi_1} X_0 =X$ for the following reason: if cycles on $X_0$ have controlled incidence with $Y_0$, it is not necessarily the case that their proper transforms via $\\pi_1$ will have suitably controlled incidence with $Y_1 \\hookrightarrow X_1$.\nFor example, suppose $Y_0$ is a linearly embedded $\\bb{P}^2$ in $X_0 = \\bb{P}^4$, and $Y_1$ is the preimage ${(\\pi_1)}^{-1}(L)$ of a line $L \\hookrightarrow Y_0$ via the blowup.\nThen $Y_1 \\to L$ is a smooth morphism between smooth varieties, and has geometrically integral fibers.  The only sensible stratification is\n$$\\emptyset = X^4 \\hookrightarrow L = X^3 \\hookrightarrow Y_0 = X^2 = X^1 \\hookrightarrow X ;$$\nsince no lines are distinguished and no line has a distinguished point, there is no canonical way to define a non-empty finite set $X^4$.  Now if $S \\hookrightarrow X$ satisfies $\\dim (S \\cap X^2) = \\dim (S \\cap X^3) = 0$, then $S$ satisfies the perversity condition $ \\overline p = (0, 1, 1)$ (and hence also the condition $\\overline q = (1,1,2)$).  Therefore the pair $(S,S)$ satisfies a pair of complementary perversity conditions.   However, the proper transform $S_1 \\hookrightarrow X_1$ satisfies $\\dim (S_1 \\cap Y_1) = 1$, so that $(S_1, S_1)$ violates the condition $p + q \\leq 1$ for the morphism $\\pi_2$ required by Proposition \\ref{smooth case}.\n\nThe self-intersection $S \\cdot S$ agrees with $\\pi_* (S_2 \\cdot S_2)$,\nbut the ``reason\" involves a feature of the morphism $X_2 \\to X_0$ which cannot be seen from its constituent factors: the ``error term\" $\\pi^* (S) - S_2$ is represented by surfaces supported over the finite set $S \\cap X^3$, and these can be moved away from each other (and from $S_2 \\cap \\pi^{-1}(S \\cap X^3)$) via rational functions on $L$.\nIn fact the same procedure works in the singular case, with possible modifications due to the more complicated geometry; see the penultimate paragraph of the proof of Proposition \\ref{intro smooth ex} for details.\n\\end{remark}\n\n\n\\section{Intersection with divisors} \\label{sec:divisors} In this section we construct a pairing between divisors and $r$-cycles (for $r \\geq 1$) by using a stratification obtained from a resolution of singularities.  We make use of the ring structure on $A_* (X)$ for nonsingular $X$, and we use that products respect supports in the following sense: if $A, B \\hookrightarrow  X$ are cycles of dimensions $r,s$, then $A \\cdot B \\in A_{r+s-d}( |A | \\cap | B|)$ is a well-defined cycle class \\cite[Ch.~8]{Ful}.  Often we use that $|A| \\cap |B|$ (or its image via a resolution) cannot support a cycle of the relevant dimension.\n\nWe use $(-)^i$ to indicate the intersection $|(-)| \\cap |X^i|$ in the remainder of the paper.\n\n\n\\begin{theorem} \\label{divisor pairing}\nLet $X$ be a $d$-dimensional variety over $k$, and let $\\pi : \\widetilde{X} \\to X$ be a strong resolution of singularities.  \nSuppose a stratification of $X$ satisfies (BC-1) and (AC) with respect to $\\pi$.\nLet $\\overline{p}, \\overline{q}$ be perversities such that $\\overline{p} + \\overline{q} = \\overline{t}$.\nThen the assignment\n$$(D, \\alpha) \\mapsto \\pi_* (\\widetilde{D} \\cdot \\widetilde{\\alpha})$$\ndetermines a well-defined pairing\n$$A^w_{d-1, \\overline{p}}(X)_\\mathbb{Q} \\otimes A^w_{r, \\overline{q}}(X)_\\mathbb{Q} \\to A_{r-1}(X)_\\mathbb{Q}$$\nfor any $r \\geq 1$.\n\\end{theorem}\n\n\\begin{proof} First we show the assignment is compatible with $\\overline{p}$-rational equivalences of divisors.  If $D$ and $D'$ are divisors with $D \\sim_{w, \\overline{p}} D'$, then in particular $D - D' = \\divfcn (f)$ for some $f \\in k(X)$.  Viewing instead $f \\in k (\\widetilde{X})$, we find $\\widetilde{D} - \\widetilde{D'} = e_D + \\divfcn(f)$, where the error term $e_D$ is an exceptional divisor for $\\pi$, and is supported over $(D \\cup D') \\cap X^2$.  Since $\\widetilde{\\alpha}$ is not contained in the exceptional locus, the intersection $\\widetilde{\\alpha} \\cap e_D$ is proper.\n\nSince a stratification satisfying (BC-1) refines the fiber dimension stratification, the preimage $\\pi^{-1}(T)$ of a divisor $T \\hookrightarrow X^i \\setminus X^{i+1}$ has dimension no larger than $(d-i-1)+(i-1) = d-2$.  Therefore $e_D$ is supported over the strata for which $p_i \\geq 1$.   For these strata, $q_i \\leq i-2$, so $\\dim (\\alpha^i) \\leq r -2$.  But now $\\pi_* ( e_D \\cdot \\widetilde{\\alpha})$ is an $(r-1)$-cycle supported in a subscheme of dimension $r-2$, hence $\\pi_* ( e_D \\cdot \\widetilde{\\alpha}) = 0$.\n\nThe vanishing of $\\pi_* (e_D \\cdot e_\\alpha)$ is proved by a similar argument: if $e_\\alpha \\not \\subseteq e_D$, then an identical argument applies.  If $e_\\alpha \\subseteq e_D$, then $e_\\alpha$ is supported over a subscheme of dimension at most $r-2$ (since it is supported over those strata for which $q_i \\leq i-2$), so that $e_D \\cdot e_\\alpha$ is represented by a cycle having positive generic dimension over its image.\n\nNow we show the assignment is compatible with $\\overline{q}$-rational equivalences of $r$-cycles, i.e., the vanishing $\\pi_* (\\widetilde D \\cdot e_\\alpha)= 0$.  If $\\alpha \\sim_{w, \\overline{q}} \\alpha'$, then $\\widetilde{\\alpha} - \\widetilde{\\alpha'} \\in A_r(\\widetilde{X})$ is represented by a cycle which is supported over $A \\cap X^2$, where $A$ is an $r$-cycle of perversity $\\overline{q}$.  (Namely $A$ consists of the union of the supports of the terms appearing in the equation of rational equivalence, i.e., the image via $\\pi$ of $e(\\alpha, \\alpha')$.)\n\nFirst assume $q_2 = 0$, so $\\alpha$ and $\\alpha'$ meet the singular locus $X^2$ properly.  Then any component $e_\\alpha$ of $e(\\alpha, \\alpha)$ is an $r$-cycle having generic dimension at least $2$ over its image, and this image is contained in the (at most) $(r-2)$-dimensional set $A \\cap X^2$.  Therefore $ \\widetilde D \\cdot e_\\alpha$ is represented by an $(r-1)$-cycle having generic dimension at least $1$ over its image, and so $\\pi_* ( \\widetilde D \\cdot e_\\alpha) =0$.\n\nFinally assume $p_2 = 0$; this is the most interesting case.  We may assume $e := e_\\alpha$ is integral and first appears on $X_{n+1} = \\Bl_{C_n} X_n$.  Let $\\overline{e} \\hookrightarrow X_{n+1}$ denote the ($r$-dimensional) image of $e$, and let $e_n \\hookrightarrow C_n \\hookrightarrow X_n$ denote the image of $e$.  Since $e$ first appears on $X_{n+1}$, there is an exceptional divisor $E'$ for the morphism $X_{n+1} \\to X_n$ which contains $\\overline e$ and dominates $C_n$.  Since $e \\to e_n$ is birational, there exists a component $E \\subseteq \\widetilde E \\hookrightarrow \\widetilde X$ which contains $e$ and maps birationally onto $E'$.  (Since  the exceptional locus of $X_{n+1} \\to X_n$ is flat over $C_n$, every exceptional component of $X_{n+1} \\to X_n$ must dominate $C_n$.)\nLet $R_n \\hookrightarrow C_n$ denote the locus determined by the reducible fibers over $C_n$ as described in condition (AC) in Definition $\\ref{strat res defn}$; we will use that $R_n$ contains the reducible fiber locus of $E \\to C_n$.  Let $V \\hookrightarrow W$ denote the image of $R_n$ in $X$.  \n\nNote that $\\dim (e_n) \\leq r-1$, hence if $e_n$ has positive generic dimension over its image in $X^2$, then we may proceed as in the case $q_2=0$.  So we may assume $e_n$ is generically finite over its image $\\overline{e_n} \\hookrightarrow W \\hookrightarrow X$.  Here $W$ denotes the image of $C_n$; say $\\dim W = d-c$, so that $W$ is a component of $X^c \\hookrightarrow X^2$.  We have $\\dim ( \\overline{e_n}) = r-1 = (r-c) + (c-1)$, hence $q_c = c-1$ and $p_c = 0$, so $D$ meets $W$ properly.\n\nLet $D_n \\hookrightarrow X_n$ denote the proper transform of $D$ via $X_n \\to X$, and $\\widetilde{D} \\hookrightarrow \\widetilde{X}$ its proper transform via $\\widetilde{X} \\to X$.  Since $D$ does not contain $W$, $D_n$ does not contain $C_n$ and $\\widetilde{D}$ does not contain $E$.\nThe situation is summarized by the following commutative diagram:\n\n\n$$\\xymatrix{ & e \\ar[r] \\ar[d] & E \\ar[r]^-i \\ar[d]_{\\pi _E } & \\widetilde{X} \\ar[d]^-\\pi  & \\widetilde{D} \\ar[l] \\ar[d] \\\\\nR_n \\ar@\/_1pc\/[rr]|\\hole  \\ar[d] & e_n \\ar[r] \\ar[d] & C_n \\ar[d] \\ar[r]^-{i_n}  & X_n \\ar[d] & D_n \\ar[l] \\ar[d] \\\\\nV \\ar@\/_1pc\/[rr] & \\overline{e_n} \\ar[r] &  W \\ar[r] & X & D \\ar[l]  }$$\n\n\\medskip\n\nWe assume that $R_n$ is not contained in $D_n$, and later we handle the case $D_n \\supseteq R_n$ by a separate argument.  \nA priori we have  $\\pi_E^{-1} |D_n \\cap C_n| \\supseteq | \\widetilde{D}  \\cap E |$, but $D_n \\not \\supset R_n$, together with the condition on fiber integrality, implies the generic points of $\\pi_E^{-1} |D_n \\cap C_n|$ coincide with those of $|\\widetilde{D} \\cap E|$. \nSince $C_n$ is smooth, $D_n \\cap C_n$ is the support of a Cartier divisor in $C_n$.  Set $M := \\mathcal{O}_{C_n} (|D_n \\cap C_n|) \\in \\pic(C_n)$, and $L := \\mathcal{O}_{\\widetilde{X}} (\\widetilde{D})  \\in \\pic(\\widetilde{X})$.  The coincidence of the generic points implies $i^*L$ and ${\\pi_E}^* M$ are rational multiples of one another in $\\pic (E)_\\mathbb{Q}$; say $m \\cdot {\\pi_E}^* M = i^* L$ for some $m \\in \\mathbb{Q}$.\n\nSince ${\\pi_E}_* (e) = 0$, the projection formula for $\\pi_E$ implies\n\\begin{equation} \\label{push 0} {\\pi_E}_* (c_1 ( {\\pi_E}^*(M)) \\cdot e) = c_1(M) \\cdot {\\pi_E}_*(e) = 0. \\end{equation}\nThe projection formula for $i$ implies the equality of cycle classes\n\\begin{equation} \\label{proj i} i_* ( c_1 ( i^* L ) \\cdot e) = c_1(L ) \\cdot e \\in A_{r-1}(\\widetilde{X}). \\end{equation}\nThe functoriality of proper push-forward and the relation in $\\pic (E)_\\mathbb{Q}$ imply\n\\begin{equation} \\label{last one} \\pi_* i_* ( c_1 ( i^* L ) \\cdot e)  = {i_n}_* {\\pi_E}_*  ( c_1 ( i^* L ) \\cdot e) =  m \\cdot {i_n}_* {\\pi_E}_*  ( c_1({ \\pi_E}^* ( M )) \\cdot e). \\end{equation}\nThese equations together imply the vanishing $\\pi_* ( c_1(L) \\cdot e) = 0$ in $A_{r-1}(X)_\\mathbb{Q}$.\n\nWe return to the case in which $D_n \\supseteq R_n$.  Then $D$ contains the image $V$ of $R_n$ in $X$, and $V$ is a component of $X^{d-c+m}$ (for some $m \\geq 1$, since $p_c=0$).  Then $p_{d-c+m} \\geq 1$, hence $q_{d-c+m} \\leq d-c+m -2$ and therefore $\\dim (A \\cap V) \\leq r-2$.\nWe have a commutative diagram relating push-forward and pull-back\n$$\\xymatrix{A_{r-1} (\\widetilde{X}) \\ar[d] \\ar[r]^-{\\pi_*} & A_{r-1}(X) \\ar[d]^-\\cong \\\\\nA_{r-1} (\\widetilde{X} \\setminus \\pi^{-1} | A \\cap V| ) \\ar[r] & A_{r-1} (X \\setminus | A \\cap V|) }$$\nso it suffices to show $\\pi_* ( \\widetilde{D} \\cdot e) $ vanishes upon restriction to $X \\setminus | A \\cap V|$.  \nAway from $V$, the generic points of $\\pi_E^{-1} |D_n \\cap C_n|$ coincide with those of $|\\widetilde{D} \\cap E|$ (as in the case $R_n \\not \\subset D_n$), and the vanishing of $\\pi_* (\\widetilde D \\cdot e)$ in $A_{r-1}(X)$ follows.\n\\end{proof}\n\\begin{remark} Outside of the case $p_2=0$, one can work with the coarser fiber dimension stratification, and with integral coefficients.  \\end{remark}\n\n\\section{Intersections on a variety with one-dimensional singular locus} \\label{sec:onedimsing}\n\nIn this section we show a resolution may be used to defined intersection pairings on a variety with one-dimensional singular locus.  At one place we need the generic smoothness of a morphism between smooth integral $k$-schemes, so our results in this section are aimed at the case $\\charct k = 0$, though they apply to special situations in positive characteristic.  We do not require a strong resolution, but we require the smoothness of the components of the exceptional locus $\\widetilde E \\hookrightarrow \\widetilde X$.  Resolutions as in Theorem $\\ref{one dim sing}$ exist by \\cite[Thm.~1.6(2)]{BierMil}, or, since we do not use the smoothness of the centers, \\cite{Kollar:res}.\n\n\\medskip\n\\textbf{The stratification.} Let $X$ be a variety over $k$.  Suppose $\\pi : \\widetilde X \\to X$ is a resolution of the singularities of $X$ such that all exceptional components are smooth over $k$, and generically smooth over their images in $X$.  We set $X^{d-1}$ equal to the singular locus of $X$, and we define $X^d \\hookrightarrow X^{d-1}$ to be the smallest set with the following properties:\n\\begin{enumerate}\n\\item $X^d$ contains the singularities of $X^{d-1}$, i.e., contains the singularities of each component of $X^{d-1}$, and contains the points at which the components intersect;\n\\item $X^d$ contains the image of every exceptional divisor $E \\subseteq \\widetilde E \\hookrightarrow \\widetilde X$ that is contracted to a point by the resolution $\\pi : \\widetilde X \\to X$; and\n\\item for every exceptional divisor $E$ with 1-dimensional image $E_1$, suppose the morphism $E \\to E_1$ has smooth generic fiber (e.g., $\\charct k =0$); then $X^d$ contains the image of the singular fibers of $E \\to E_1$.\n\\end{enumerate}\n\n\\begin{remark} \nCondition (1) is a mild strengthening of condition (BC-3) of Definition \\ref{strat res defn}, since we additionally take into account the incidences of the components of $X^{d-1}$.  Condition (BC-1) implies condition (2) must be satisfied.  Condition (3) above is a strengthening of condition (AC).\n\\end{remark}\n\n\\begin{theorem} \\label{one dim sing} Suppose $X$ is a variety over a field $k$ such that $\\dim(X_{sing}) =1$, and suppose $\\pi : \\widetilde X \\to X$ is a resolution of singularities such that the exceptional components are \n\\begin{itemize}\n\\item$k$-smooth, and \n\\item generically smooth over their images in $X$.\n\\end{itemize}\nLet $\\overline p, \\overline q$ be perversities such that $\\overline p + \\overline q = \\overline t$.\nWith respect to the stratification defined above, the assignment $\\alpha, \\beta \\mapsto \\pi_* ( \\widetilde \\alpha \\cdot \\widetilde \\beta)$ determines a well-defined pairing $A^w_{r, \\overline p}(X) \\otimes A^w_{s, \\overline q}(X) \\to A_{r+s-d} (X)$.  \\end{theorem}\n\nAs a matter of notation, we mostly let $\\alpha$ denote the factor in which error terms are considered, so that $e_\\alpha \\in Z_*(\\widetilde X)$ is a cycle which arises by taking the proper transform of an equivalence (respecting some perversity condition) relating (say) $\\alpha$ to $\\alpha'$ on $X$.\n\n\\begin{proof} The claim is obvious if $r+s-d \\geq 2$, for then $e_\\alpha \\cdot \\widetilde \\beta, \\widetilde \\alpha \\cdot e_\\beta,$ and $e_\\alpha \\cdot e_\\beta$ are all represented by cycles of dimension $\\geq 2$ supported over $X^{d-1}$, hence have generic dimension $\\geq 1$ over their images, so that all vanish after $\\pi_*$ is applied.  This case requires no perversity condition at all.\n\nConsider the case $r+s-d=1$.  If the error term $e_\\alpha$ is supported over a finite set, then $e_\\alpha \\cdot \\widetilde \\beta$ is represented by a 1-cycle which is contracted to a finite set by $\\pi$ (since $e_\\alpha$ is so contracted), hence $\\pi_* (e_\\alpha \\cdot \\widetilde \\beta) = 0$.  The same argument works with $e_\\beta$ in the place of $\\widetilde \\beta$.  Therefore we may assume $\\dim (\\alpha^{d-1}) =1$, so that $p_{d-1} = d - r$ and $q_{d-1} \\leq r-2$, and hence $\\dim (\\beta^{d-1})  \\leq 0$.  Working one component at a time, we may assume $e_\\alpha$ is contained in a single (smooth) exceptional component $i :E \\hookrightarrow \\widetilde X$.  Now $e_\\alpha \\cdot \\widetilde \\beta = i_* (e_\\alpha \\cdot i^* (\\widetilde \\beta))$, and $i^* (\\widetilde \\beta)$ is represented by an $(s-1)$-dimensional cycle supported over the finite set $\\beta^{d-1}$.  Therefore $e_\\alpha \\cdot i^* (\\widetilde \\beta)$ is represented by a 1-cycle which is contracted to a finite set, so $ ( \\pi \\circ i)_* ( e_\\alpha \\cdot i^* (\\widetilde \\beta) ) = 0$.  Therefore $\\pi_* (e_\\alpha \\cdot \\widetilde \\beta) = 0$, as desired.  Since one of $e_\\alpha, e_\\beta$ must be supported over a finite set in $X^{d-1}$, the term $\\pi_* (e_\\alpha \\cdot e_\\beta)$ vanishes for the same reason.\n\nNow we consider the case $r+s-d=0$.  Now $\\dim (\\alpha^{d-1}) =1$ implies $\\dim (\\beta^{d-1}) < 0$, so the interesting situation is when both $\\alpha^{d-1}$ and $\\beta^{d-1}$ are finite; only one of $\\alpha, \\beta$ is allowed to meet $X^d$.  As in the previous paragraph, we choose a (smooth) exceptional component $i : E \\hookrightarrow \\widetilde X$ containing $e_\\alpha$.  Both $e_\\alpha$ and $i^* (\\widetilde \\beta)$ are supported over finite sets in $X^{d-1}$, and their incidence must occur over points in $X^{d-1} \\setminus X^d$.  \nAgain working one component at a time, the cycles $e_\\alpha$ and $i^* (\\widetilde \\beta)$ are disjoint unless they are supported over the same point $x \\in X^{d-1} \\setminus X^d$.  Let $j : E_x \\hookrightarrow E$ denote the inclusion of the exceptional fiber over $x$.  By construction of the stratification, $E_x$ is smooth.  \n\nSet $\\beta' := i^* (\\widetilde \\beta)$, and let $N$ denote the normal bundle of the embedding $j$.  Using the projection formula and the self-intersection formula (\\cite[Cor.~6.3]{Ful}), we find:\n$${j}_* (e_\\alpha) \\cdot_E  {j}_* ( \\beta' ) = j_* (e _\\alpha \\cdot_{E_x} j^* j_* \\beta' ) = j_* (e_\\alpha \\cdot_{E_x} ( c_1 (N) \\cap \\beta')).$$\nBut $E_x$ is principal, hence $c_1 (N) = 0$, and therefore\n$${j}_* (e_\\alpha) \\cdot_E  {j}_* (\\beta') = e_\\alpha \\cdot_ E i^* (\\widetilde \\beta) =0.$$\nThe projection formula for $i$ implies $e_\\alpha \\cdot \\widetilde \\beta = 0 \\in A_0( \\widetilde X)$.\n\nThe vanishing of $e_\\alpha \\cdot e_\\beta$ holds for a similar reason: the nontrivial case is when both $e_\\alpha$ and $e_\\beta$ are supported over the same point $x \\in X^{d-1} \\setminus X^d$.  But then\n$j_* (e _\\alpha) \\cdot_E j_* (e_\\beta) = j_* ( e_\\alpha \\cdot_{E_x}  ( c_1(N) \\cap e_\\beta ) )$, and the vanishing of $c_1(N)$ allows us to conclude.   \\end{proof}\n\n\\begin{remark}\nThe basic obstacle to extending the above analysis to the case $\\dim(X_{sing})=2$ is that the singular fibers of $E \\to E_2$ (here $E$ is a component of the singular locus with 2-dimensional image $E_2 \\hookrightarrow X$) may be supported over a divisor in $E_2$, and, in the case $\\alpha^{d-1}$ and $\\beta^{d-1}$ are both finite, the perversity conditions do not rule out the incidence being supported in a singular fiber.\n\\end{remark}\n\n\n\\section{2-cycles on a fourfold} \\label{sec:4fold} In this section we work on a normal quasi-projective $4$-dimensional variety $X$; more precisely we assume the singularities of $X$ occur in codimension at least $2$.  As in the previous cases, the resolutions we require exist in characteristic zero.\nWe describe more explicitly the construction of the stratification satisfying (BC-1), (BC-2), (BC-3), and (AC) with respect to a strong resolution.  We start by declaring $X^2$ is the singular locus of $X$.\n\n\\begin{enumerate}\n\n\\item (BC-1) For every (smooth, integral) two-dimensional center $S$ such that the composition $S \\to X^2$ is dominant over a component of $X^2$, the image of the positive-dimensional (i.e., one-dimensional) fibers of $S \\to X^2$ is a finite set in $X^2$.  Place this set in $X^4$.\n\n\\item For every two-dimensional center $S$ such that the composition $S \\to X^2$ has one-dimensional image $W \\hookrightarrow X^2$,\n\n\\subitem (BC-1) place $W$ in $X^3$, and\n\n\\subitem (BC-3) place the singularities of $W$ in $X^4$.\n\n\\subitem (BC-2)  The morphism $p : S \\to W$ has integral generic fiber.  Place the zero-dimensional set $ \\{ w \\in W | p^{-1}(w) \\text{ is reducible} \\}$ in $X^4$.\n \n\\item (BC-1) If the two-dimensional center $S$ of a blowup has zero-dimensional image in $X^2$ (so it lies over a single point $x \\in X^2$), then $x$ is placed in $X^4$.\n\n\\item (AC) Let $p : \\widetilde{E}_{\\text{dom }S} \\to S$ denote the canonical morphism.  Suppose $\\widetilde{E}_{\\text{dom }S}$ has $t$ components.  Then the image in $X^2$ of the closed set $\\{ s \\in S | p^{-1}(s) \\text{ has more than $t$ components} \\}$\nis placed in $X^3$.\n\\item For every one-dimensional center $C$ which is generically finite onto its image in $X^2$,\n\n\\subitem (BC-1) place the image curve in $X^3$, and\n\n\\subitem (BC-3) place its singularities in $X^4$.\n\n\\subitem (BC-1) If a one-dimensional center $C$ has zero-dimensional image in $X^2$ (so it lies over a single point $x \\in X^2$), then $x$ is placed in $X^4$.\n\n\\subitem (AC) Place in $X^4$ the closed set in $C$ over which $\\widetilde{E}_{\\text{dom }C} \\to C$ has more components than does $\\widetilde{E}_{\\text{dom }C}$ itself.\n\n\\item (BC-1) The image in $X^2$ of every zero-dimensional center $Z$ is placed in $X^4$.\n\n\\end{enumerate}\n\nThe instances of condition (BC-1) are necessary to guarantee the stratification refines the stratification by fiber dimension.  In the course of the proof we point out where the other conditions are used; the condition (BC-3) in (5) does not seem to be necessary, but (BC-3) in (2) is used.\n\nWe will use notation from the following diagram.\n\n$$\\xymatrix{ \\widetilde{X} \\ar[r] & \\Bl_{Z_n}X_n \\ar[r]  & X_n \\ar[r] & \\ldots \\ar[r] & X \\\\\n\\widetilde{E} \\ar[u] & E_n \\ar[u] \\ar[r]^-{p_n} & Z_n \\ar[u] & \\ldots & X^2 \\ar[u] \\\\ }$$\nThe center $Z_n$ will be written as $S_n$ when it is two-dimensional and as $C_n$ when it is one-dimensional.  Note that each $Z_n$ is smooth, and each $p_n : E_n \\to Z_n$ is a flat morphism.  Zero-dimensional centers play no essential role since incidences in $X^4$ are forbidden by the perversity condition.\nBlowups along three-dimensional centers are finite morphisms (by normal flatness) and do not influence the stratification.\n\n\n\\begin{theorem} \\label{4fold} Let $X$ be a quasi-projective fourfold over $k$, and let $\\pi : \\widetilde X \\to X$ be a strong resolution of singularities.\nSuppose a stratification of $X$ satisfies (BC-1), (BC-2), (BC-3), and (AC) with respect to $\\pi$.\nLet $\\overline{p}, \\overline{q}$ be perversities such that $\\overline{p} + \\overline{q} = \\overline{t}$.  The assignment $(\\alpha, \\beta) \\mapsto \\pi_* (\\widetilde{\\alpha} \\cdot \\widetilde{\\beta})$ determines a well-defined pairing $A^w_{2, \\overline{p}} (X)_\\mathbb{Q} \\otimes A^w_{2, \\overline{q}} (X)_\\mathbb{Q} \\to A_0(X)_\\mathbb{Q}.$\n\\end{theorem}\n \n\nThere are essentially three complementary pairs of perversities to analyze, and these are handled in the next three propositions.  For each pair the strategy is to consider possible locations of the generic points of error terms, and for each location we move the error term away from the other 2-cycle.  If the error term first appears on the blowup of $X_n$ along $Z_n$, then the move is achieved by finding a suitable rational function on $Z_n$.  \n\n\\begin{proposition} Theorem \\ref{4fold} is true for $\\overline{p}=\\overline{0}, \\overline{q} = \\overline{t}$; and for $\\overline{p}=(0,0,1), \\overline{q}=(1,2,2)$. \\end{proposition}\n\n\\begin{proof} If $\\alpha \\sim_{w, \\overline{p}} \\alpha'$ and $z$ is a cycle appearing in the equation relating $\\alpha$ to $\\alpha'$, the dimension of $\\pi^{-1}(z^i)$ is at most $2-i + (i-1) =1$.  Since the preimage of the exceptional part is $1$-dimensional, it cannot support a $2$-cycle and the error terms vanish.  Therefore $\\widetilde{\\alpha} \\sim \\widetilde{\\alpha'}$ and clearly then $\\pi_* (\\widetilde{\\alpha} \\cdot \\widetilde{\\beta}) \\sim \\pi_* (\\widetilde{\\alpha'} \\cdot \\widetilde{\\beta})$.\n\nIt remains to check the compatibility with $\\overline{q}$-rational equivalence in $\\beta$, i.e., the vanishing $\\pi_* (\\widetilde{\\alpha} \\cdot e_\\beta) =0$, where $e_\\beta$ is a component of the error term $e(\\beta, \\beta')$.  Now $\\alpha^2$ is a finite set contained in the smooth part $X^2 \\setminus X^3$ of $X^2$, and $\\alpha^3$ is empty.  If the generic point of $e_\\beta$ lies over $X^3$, then $\\widetilde{\\alpha} \\cap e_\\beta = \\emptyset$ and we are done.\n\nTherefore we suppose the generic point of $e_\\beta$ lies over $X^2 \\setminus X^3$, and let $E \\subseteq \\widetilde{E} \\hookrightarrow \\widetilde{X}$ be a component of the exceptional locus which contains $e_\\beta$.  If the image of $E$ has dimension less than or equal to $1$, then $E$ has generic dimension at least $2$ over its image, contradicting the assumption that the generic point of $e_\\beta$ lies over $X^2 \\setminus X^3$.  Therefore $e_\\beta$ first appears on some blowup $\\Bl_{S_n}X_n$ where $S_n$ is a smooth surface on some intermediate variety $X_n$.  Now let $E$ denote an exceptional divisor which contains $e_\\beta$ (as above), and maps birationally onto a divisor which is exceptional for the morphism $\\Bl_{S_n}X_n \\to X_n$ (i.e., $E$ also first appears on $\\Bl_{S_n}X_n$).\n\n Let $a : S_n \\to X^2$ denote the canonical morphism.  Note that ${a}^{-1}(\\alpha^2)$ is finite by our construction of the stratification.\nLet $e_n$ denote the (one-dimensional) image of $e_\\beta$ in $S_n$.  There exists a $1$-cycle $C$ on $S_n$ which is rationally equivalent to $e_n$, and with the property that $C \\cap {a}^{-1}(\\alpha^2) = \\emptyset$.  In other words, there is a rational function $g_n: S_n \\dashrightarrow \\mathbb{P}^1$ such that $[g_n(0)] = e_n + Z$, and such that both $Z$ and $[g_n(\\infty)]$ are disjoint from ${a}^{-1}(\\alpha^2)$.\n\nNow consider the composition $g: E \\to S_n \\dashrightarrow \\mathbb{P}^1$.  By the fiber integrality hypothesis (i.e., since (AC) in (4) places the image of the reducible fibers of $E \\to S_n$ into $X^3$), the support of $e_\\beta$ coincides (set-theoretically) with the preimage of $e_n$ by the morphism $E \\to S_n$.  \nTherefore $g$ provides a rational equivalence between $m \\cdot e_\\beta + Z'$ and $Z''$, where $Z'$ is supported over $Z$, and $Z''$ is supported over $[g_n(\\infty)]$.  Hence both $Z'$ and $Z''$ are disjoint from $\\widetilde{\\alpha}$.  Therefore $\\widetilde{\\alpha} \\cdot (m \\cdot e_\\beta) \\sim \\widetilde{\\alpha} \\cdot (Z'' - Z') = 0$ and so $\\widetilde{\\alpha} \\cdot e_\\beta$ is zero in $A_0(\\widetilde{X})_\\mathbb{Q}$, as desired.\n\nAn identical argument handles the pair $\\overline{p}=(0,0,1), \\overline{q}=(1,2,2)$ since it imposes the same conditions on $\\alpha$ and $\\beta$. \\end{proof}\n \n\\begin{proposition}  \\label{intro smooth ex} Theorem \\ref{4fold} is true for $\\overline{p}=(0,1,1), \\overline{q} = (1,1,2)$. \\end{proposition}\n\n\\begin{proof}  In this case $\\alpha^2$ is finite and $\\alpha^4$ is empty, hence if $e(\\alpha, \\alpha')$ is nonempty it consists of several surfaces lying over some zero-cycle $Z \\hookrightarrow X^3 \\setminus X^4$.  Thus a component $e_\\alpha$ of $e(\\alpha, \\alpha')$ is an irreducible surface which first appears on $\\Bl_{C_n}X_n$, the blowup of some intermediate variety $X_n$ along a smooth one-dimensional center $C_n$.  Let $E$ denote an exceptional divisor that contains $e_\\alpha$ and first appears when $e_\\alpha$ does (so that $E \\hookrightarrow \\widetilde{E}_{C_n}$).\nThe image of $e_\\alpha$ is a point $c \\in C_n$; since $C_n$ is finite over its image in $X^2$, the proper transform of $\\beta$ via $X_n \\to X$ meets $C_n$ in a finite set.\n\nThere exists a rational function $g_n: C_n \\to \\mathbb{P}^1$ such that $[g_n(0)] = c + Z$, and such that both $Z$ and $[g_n(\\infty)]$ are disjoint from the proper transform of $\\beta$.  Consider the rational function $g: E \\to C_n \\to \\mathbb{P}^1$.  Since the image of $c \\in C_n$ lies in $X^3 \\setminus X^4$, the fiber of $E \\to C_n$ over $c$ is irreducible (by (AC) in (5)), and therefore $e_\\alpha$ coincides set-theoretically with $E \\cap \\pi^{-1}(c)$.\nTherefore $g$ provides a rational equivalence between $m \\cdot e_\\alpha + Z'$ and $Z''$, where $Z'$ is supported over $Z \\hookrightarrow X^3 \\setminus X^4$, and $Z''$ is supported over $[g_n(\\infty)]$.  Consequently $e_\\alpha \\cdot \\widetilde{\\beta} = 0$ in $A_0(\\widetilde{X})_\\mathbb{Q}$.   Note $e_\\beta$ must be supported over a finite set in $C_n$, so the same construction moves $e_\\alpha$ away from $e_\\beta$.\n\nNow we show the vanishing of $\\widetilde{\\alpha} \\cdot e(\\beta, \\beta')$.  First we consider components $e_\\beta$ of $e(\\beta, \\beta')$ whose generic points lie over $X^2 \\setminus X^3$.  Then $e_\\beta$ first appears on the blowup of a two-dimensional center $S_n$ which is dominant over a component of $X^2$.  The image of $e_\\beta$ in $S_n$ is a subvariety of dimension $1$, and $e_n := p_n (e_\\beta)$ is not contracted by $a: S_n \\to X^2$.  (If $e_\\beta$ were supported over a subvariety contracted by $a$, then it would have generic dimension $2$ over its image.)  We choose an exceptional divisor $E \\supset e_\\beta$ as usual.\n\n\nSince $\\alpha^4$ is empty, ${a}^{-1}(\\alpha^2)$ is finite.  Therefore $e_n$ is rationally equivalent to a $1$-cycle on $S_n$ which is disjoint from ${a}^{-1}(\\alpha^2)$, i.e., there exists a rational function $g_n: S_n \\dashrightarrow \\mathbb{P}^1$ such that $[g_n(0)] = e_n + Z'$, and such that both $Z'$ and $[g_n(\\infty)]$ are disjoint from ${a}^{-1}(\\alpha^2)$.  Now we consider the composition $g := g_n \\circ p_n : E \\to S_n \\dashrightarrow \\mathbb{P}^1$.  By the integrality condition on the fibers (condition (AC) in (4)), $[g(0)]$ is a multiple of $e_\\beta$ plus a 2-cycle supported over $Z'$, and $[g(\\infty)]$ is a 2-cycle supported over $[g_n(\\infty)]$.  In particular, some multiple of $e_\\beta$ is rationally equivalent to a cycle disjoint from $\\widetilde{\\alpha}$.\n\n\nNext we consider error terms $e_\\beta$ whose generic points are supported over $X^3 \\setminus X^4$.  There are two possibilities for first appearance.\nSuch a term may first appear on the blowup along a (smooth) one-dimensional center $C_n$ (with one-dimensional image in $X^2$); \nthis is handled by finding a rational function on $C_n$ which, upon precomposing with a canonical morphism from an exceptional divisor to $C_n$, gives a rational equivalence between some multiple of $e_\\beta$ and a $2$-cycle which is disjoint from the preimage of $\\alpha^3$ (as $\\alpha^3$ necessarily meets $C_n$ in a finite set) and hence from $\\widetilde{\\alpha}$.\n\nThe other possibility is that the error term $e_\\beta$ (with generic points supported over $X^3 \\setminus X^4$) first appears on the blowup along a (smooth) two-dimensional center $S_n$, in which case the one-dimensional image of $e_\\beta$ inside $S_n$ is contracted to a point by the morphism $S_n \\to X^2$: either the center $S_n$ has one-dimensional image in $X^2$, or $e_\\beta$ is supported over some exceptional part of the generically finite morphism $S_n \\to X^2$.  But $e_\\beta$ is exceptional for $S_n \\to X^2$ implies the generic point of $e_\\beta$ is supported over $X^4$, so we may assume the two-dimensional center $S_n$ has one-dimensional image $W \\hookrightarrow X^2$.  By our definition of $X^4$, (the finite sets) $\\alpha^3 \\cap W$ and $\\beta^3 \\cap W$ are supported in the smooth locus of $W$, and in the locus over which $S_n \\to W$ has integral fibers (by (BC-2) and (BC-3) as in (2)).  Let $q$ denote the canonical morphism $E \\to S_n \\to W$, and let $w \\in W$ be the image of $e_\\beta$ via $q$.  Note that $e_\\beta$ coincides set-theoretically with $q^{-1}(w)$.  There exists a rational function $g: W \\dashrightarrow \\mathbb{P}^1$ such that $[g(0)] = w + Z$, and such that both $Z$ and $[g(\\infty)]$ are disjoint from $\\alpha^3$.  Then the rational function $E \\to S_n \\to W \\xrightarrow{g} \\mathbb{P}^1$ provides a rational equivalence between some multiple of $e_\\beta$ and a cycle which is disjoint from the preimage of $\\alpha^3$, so that $\\widetilde{\\alpha} \\cdot e_\\beta \\sim 0$ as desired.\n\nFinally, components $e_\\beta$ of $e(\\beta, \\beta')$ supported over $X^4$ are automatically disjoint from $\\widetilde{\\alpha}$.  In particular this applies if $e_\\beta$ is supported over some exceptional part of the generically finite morphism $S_n \\to X^2$. \\end{proof}\n\n\\begin{proposition} Theorem \\ref{4fold} is true for $\\overline{p}=(0,1,2), \\overline{q} = (1,1,1)$. \\end{proposition}\n\n\\begin{proof} In this case $\\alpha^2$ is finite (but may have support in $X^4$), $\\beta^2$ is $1$-dimensional, $\\beta^3$ is finite, and $\\beta^4$ is empty.  The error terms appearing in $e(\\alpha, \\alpha')$ cannot be supported over $X^2 \\setminus X^3$ since $p_2=0$.  The error terms supported over $X^3 \\setminus X^4$ are handled (moved away from the finite set $\\beta^3$) as in the previous case.  The error terms supported over $X^4$ are disjoint from $\\widetilde{\\beta}$ since $\\beta^4 = \\emptyset$.  The same reasoning applies with $\\widetilde \\beta$ replaced by $e_\\beta$.\n\nNow we show the vanishing $\\pi_* (\\widetilde \\alpha \\cdot e_\\beta) =0$.\nThe error terms in $e(\\beta, \\beta')$ lying over $X^3$ are handled as in the previous case.  It remains to show $\\widetilde{\\alpha} \\cdot e_\\beta = 0$ when $e_\\beta$ is a component of $e(\\beta, \\beta')$ whose generic point is supported over $X^2 \\setminus X^3$.  We let $e_0$ denote the image of $e_\\beta$ in $X^2$, and let $X^2_e$ denote the component of $X^2$ that contains $e_0$.\n\nLet $S$ denote the first two-dimensional center that dominates $X^2_e$.  Then $S \\to X^2_e$ is a resolution of singularities, and it (Stein) factors as $S \\to S' \\to X^2_e$, where $b: S' \\to X^2_e$ omits those blowups along centers landing in $X^4$.  (While $S$ is a closed subvariety in some intermediate $X_j$, the variety $S'$ may not admit a closed immersion into any $X_j$.)  Thus $b$ is a finite morphism, the singular set of $S'$ is contained in $b^{-1}(X^4)$, and the proper transform $e_0^{'} \\hookrightarrow S'$ of $e_0$ via $S' \\to X^2_e$ is supported in the smooth locus of $S'$.\n\nSuppose $e_\\beta$ first appears on the blowup of the two-dimensional center $S_n$, and consider the Stein factorization $S_n \\to S_n' \\to S'$ of the morphism $S_n \\to S'$. (Note $S_n \\to S$ is dominant since the generic point of $e_\\beta$ is supported over $X^2 \\setminus X^3$, and of course $S_n=S$ is possible.)   Again choose an exceptional component $E \\supset e_\\beta$ dominating $S_n$.  Let $e_n$ denote the image of $e_\\beta$ in $S_n$, and $e_n'$ its image in $S_n'$.  Since $S_n' \\to S' \\to X^2_e$ is finite, the preimage of $X^4$ in $S_n'$ is finite.\n\nThe morphism $S_n \\to S_n'$ contracts exactly those curves lying over curves contracted by $S \\to S'$.  Since the curves contracted by $S \\to S'$ (more precisely the zero-dimensional image in $X^2$ of such curves) are disjoint from $e_0$, the morphism $S_n \\to S_n'$ is an isomorphism in a neighborhood of $e_n$, hence $S_n'$ is smooth in a neighborhood of $e_n'$.  Now we proceed as usual.  We find a rational function on $S_n'$ moving $e_n'$ away from the finite preimage of $X^4$.  Then the composite rational function $E \\to S_n \\to S_n' \\dashrightarrow \\mathbb{P}^1$ moves a multiple of $e_\\beta$ away from $\\widetilde{\\alpha}$, as required.\n\\end{proof}\n\n\\section{Necessity of condition (AC)} \\label{need integral fibers}\n\nSince the arguments of the previous section may seem intricate, one may ask if there is a simpler description of a stratification for which intersecting on the resolution induces a well-defined pairing.  In this section we show the stratification according to the fiber dimension of the resolution is insufficiently fine to obtain a well-defined intersection product, even when all the strata are smooth.  The example applies to both equivalence relations $\\sim_{\\overline p}$ and $\\sim_{w, \\overline p}$.\nWe work over a field $k$ of characteristic $\\neq 2$.\n\n\\medskip\n\\textbf{An affine example.}  Let $X \\hookrightarrow \\mathbb{A}^4$ be the hypersurface defined by the vanishing of $x^2 - y^2 + tz^2$.  Then $X$ is singular along the line $L$ defined by $x=y=z=0$, and we claim a resolution $\\pi : \\widetilde{X} \\to X$ is obtained by blowing up $X$ along $L$.  On the patch where $x$ generates (say $y = y'x$ and $z = z'x$), the blowup is defined by the vanishing of $1- {(y')}^2 + t{(z')}^2$, which is smooth.  On the patch where $y$ generates, the blowup is defined by the vanishing of ${(x')}^2 -1 + t{(z')}^2$, which is also smooth.  On the patch where $z$ generates, the blowup is defined by the vanishing of ${(x')}^2 - {(y')}^2 +t$, and this too is smooth.\n\nNote that all of the fibers of $\\pi^{-1}(L) \\to L$ are one-dimensional.  Since $L$ is regular, this implies $\\pi^{-1}(L) \\to L$ is flat and so $L \\hookrightarrow X$ is normally flat.  Taking into account only the fiber dimensions in the resolution and the singularities of the centers, we are led to the stratification $X^3 = \\emptyset \\hookrightarrow X^2 = X^1 = L \\hookrightarrow X$.  The relevant feature (which this stratification ignores) is that the fiber of $\\pi$ over $(0,0,0,0)$ has two components ($x'+y' = 0$ and $x'-y' =0$), whereas the fiber over $(t_0, 0, 0, 0)$ is irreducible for $t_0 \\neq 0$.\n\n\nLet $D \\hookrightarrow X$ be the divisor defined by the ideal $(x+y,t)$, let $\\alpha_0$ be the $1$-cycle defined by $(x-y,t,z)$, and let $\\alpha_1$ be the $1$-cycle defined by $(x-y,t-1,z)$.  Each of the cycles $D, \\alpha_0$, and $\\alpha_1$ meets $L$ in a finite set, so $D$ has perversity $\\overline 0$ and the $\\alpha$'s have perversity $\\overline 1 := (1,1,1)$.  Furthermore $\\alpha_0 \\sim \\alpha_1$ since the $\\alpha$'s arise as preimages of distinct values of the regular function $t$ on the surface $S$ cut out by $(x-y,z)$.  The equivalence respects the perversity condition $\\overline 1$ since $t : S \\to \\mathbb{A}^1$ maps $L$ isomorphically onto $\\mathbb{A}^1$, so that each fiber of $t$ meets $L$ exactly once.\n\nNevertheless, $\\widetilde{D} \\cap \\widetilde{\\alpha_0}$ consists of a single (reduced) point: on the patch where $z$ generates, the intersection occurs at $x' =  y' = t = z =0$.  On the other hand, $\\widetilde{D} \\cap \\widetilde{\\alpha_1} = \\emptyset$ (in fact $D \\cap \\alpha_1 = \\emptyset$). \nWe conclude that the stratification is too coarse for the resolution to determine a well-defined intersection product.\nOf course the example disappears if we use instead the stratification $X^3 = (0,0,0,0) \\hookrightarrow X^2=X^1 = L \\hookrightarrow X$, as implied by Theorem \\ref{divisor pairing}.\n\n\\medskip\n\\textbf{Behavior at infinity.}  For the reader who does not take degrees of zero-cycles on an affine variety too seriously, we now show the behavior persists on the projective closure.\n\nLet $\\underline {X} \\hookrightarrow \\bb{P}^4_{S,T,X,Y,Z}$ be the hypersurface cut out by $SX^2 - SY^2 + T Z^2$.  The singular locus of $\\underline  X$ consists of three components, each abstractly isomorphic to $\\bb{P}^1$:\n\\begin{itemize}\n\\item $\\Sigma_1 = Z(X,Y,Z)$, which is the closure of $L$ in the affine example;\n\\item $\\Sigma_2 = Z(X - Y, S, Z)$; and\n\\item $\\Sigma_3 = Z(X + Y, S, Z)$.\n\\end{itemize}\n\nThe intersection point $p = [0 : 1 : 0 : 0 : 0] = \\Sigma_1 \\cap \\Sigma_2 \\cap \\Sigma_3$ (which did not appear in the affine example) may be placed in ${\\underline X}^3$ (because the singular locus of $\\underline  X$ is itself singular there, or because the resolution has different behavior over $p$).  Our constructions will preserve any perversity condition which forbids incidence with $p$, hence will show\n$${\\underline X}^3 = p \\hookrightarrow {\\underline X}^2 = {\\underline X}^1 = \\Sigma_1 \\cup \\Sigma_2 \\cup \\Sigma_3 \\hookrightarrow \\underline X$$\nis not fine enough for the intersection product to be well-defined.  As in the affine example, ${\\underline X}^3$ must include $[1:0:0:0:0]$ as well.\n\nA resolution is obtained by blowing up $\\underline  X$ along $\\Sigma_1$, then blowing up the proper transforms $\\widetilde{\\Sigma_2}, \\widetilde{\\Sigma_3} \\hookrightarrow B_{\\Sigma_1} (\\underline  X)$; since $\\widetilde{\\Sigma_2}$ and $\\widetilde{\\Sigma_3}$ are disjoint on $B_{\\Sigma_1} (\\underline  X)$, the order is irrelevant.  Initially blowing up $p$ does not improve the situation, in the sense that one finds a copy of the original singularity on $B_p (\\underline X)$.  This is perhaps not too surprising since one finds Whitney umbrellas along $x=0$ and $y=0$, and these too are not resolved by blowing up the ``worst\" point in the singular locus.\n\n\\begin{proposition} \\label{ex strong res} The morphism\n$$B_{\\widetilde{\\Sigma_2} \\cup \\widetilde{\\Sigma_3}} (B_{\\Sigma_1} (\\underline  X ))\\to B_{\\Sigma_1} (\\underline  X) \\to \\underline  X$$\nis a strong resolution of the singularities of $\\underline  X$. \\end{proposition}\n\\begin{proof}\nFirst we verify that the blowup along $\\Sigma_1 \\cup \\Sigma_2 \\cup \\Sigma_3 - p$ resolves $\\underline  X - p$; for this calculation the order of the blowup is irrelevant.  Then we analyze the behavior near $p$.\n\nFor the blowup along $\\Sigma_1 - p$, we work on $S \\neq 0$ and recover the affine example: the blowup of $x^2 - y^2 + tz^2$ along $x=y=z=0$.  Hence $B_{\\Sigma_1 - p} (\\underline  X - p)$ is smooth above $\\Sigma_1 - p$.  The analysis of the other blowups is similar (work on $X \\neq 0$) and we omit the details.\n\nHaving analyzed the morphism away from $p$, we analyze the blowup along $\\Sigma_1$ where $T \\neq 0$, so that we are blowing up $sx^2 - sy^2 + z^2 = 0$ along $x=y=z=0$.  Where $x$ generates, the blowup is defined by $s - s {(y')}^2 + {(z')}^2 =0$, and is singular along $1-{(y')}^2 = s = z' = 0$; these two singular points are exactly where the proper transforms $\\widetilde{\\Sigma_2}, \\widetilde{\\Sigma_3}$ meet the fiber over $p$.  Where $y$ generates, the blowup is defined by $s{(x')}^2 -s + {(z')}^2 =0$, and is singular along ${(x')}^2-1=s=z'=0$ (so that we see the same two points).  Where $z$ generates, the blowup is smooth.\n\nThe blowup of $B_{\\Sigma_1}(\\underline  X)$ along $\\widetilde{\\Sigma_2}$ is covered by two charts:\n\\begin{itemize}\n\\item the blowup of $s - s{(y')}^2 + {(z')}^2 =0$ along $1-y' = s = z' = 0$, and\n\\item the blowup of $s{(x')}^2 -s + {(z')}^2 =0$ along $x'-1 = s= z' =0$;\n\\end{itemize}\nboth of these are smooth.  The blowup along $\\widetilde{\\Sigma_3}$ is similar, except one uses the centers $1+y' = s = z' = 0$, and $x'+1 = s= z' =0$ in these charts.\nAll three centers are normally flat in their ambient spaces since the exceptional divisors over them are irreducible, and the centers are regular and one-dimensional.\n\\end{proof}\n\n\n\\begin{lemma} \nThe stratification\n$${\\underline X}^3 = p \\hookrightarrow {\\underline X}^2 = {\\underline X}^1 = \\Sigma_1 \\cup \\Sigma_2 \\cup \\Sigma_3 \\hookrightarrow \\underline X$$\nsatisfies (BC-1), (BC-2), and (BC-3) with respect to the resolution of Proposition \\ref{ex strong res}.  With respect to this stratification,\n\\begin{itemize}\n\\item the divisor $\\underline  D = Z(X+Y, T) \\hookrightarrow \\underline X$ satisfies the perversity condition $\\overline 0$, and\n\\item the 1-cycles $\\underline  \\alpha_0 = Z(X-Y,T,Z)$ and $\\underline  \\alpha_1 = Z(X-Y, T-S,Z)$ satisfy the perversity condition $\\overline 1$ and are equivalent as 1-cycles of perversity $\\overline 1$.\n\\end{itemize} \nNevertheless, $\\deg (\\pi_* ( \\widetilde{\\underline  D} \\cdot \\widetilde{\\underline  \\alpha_0}) ) =1$ and $\\widetilde{\\underline  D} \\cap \\widetilde{\\underline  \\alpha_1} = \\emptyset$.\n\\end{lemma}\n\\begin{proof}\nThe first claim follows from the following incidence properties: $\\underline  D \\cap \\Sigma_1 = [ 1 : 0 : 0 : 0 : 0 ] \\ , \\ \\underline  D \\cap \\Sigma_2 = \\emptyset \\ , \\ \\underline  D \\cap \\Sigma_3 = [ 0 :0:1:1:0]$.\n\nNext we analyze the rational equivalence relating $\\underline  \\alpha_0 = Z(X-Y,T,Z)$ and $\\underline  \\alpha_1 = Z(X-Y, T-S,Z)$.  Both of these 1-cycles are lines in $\\bb{P}^2 \\cong Z(X - Y, Z) =: P \\hookrightarrow \\underline  X$.  Note $\\Sigma_1 \\cup \\Sigma_2 \\hookrightarrow P$, so we must ensure the equivalence $\\underline  \\alpha_0 \\sim \\underline  \\alpha_1$ can be chosen to respect the perversity condition.  Since $\\Sigma_3 \\cap P = p$, the component $\\Sigma_3$ poses no difficulty beyond that presented by $\\Sigma_1 \\cup \\Sigma_2$.\n\nWe claim there exists an $\\mathbb{A}^1$-relative 1-cycle $A \\hookrightarrow \\underline  X \\times \\mathbb{A}^1$ with the following properties:\n\\begin{enumerate}\n\\item $A_0 = \\underline \\alpha_0, A_1 = \\underline  \\alpha_1$;\n\\item all of the fibers $A_t$ are disjoint from $p$; and\n\\item all of the fibers $A_t$ meet $\\Sigma_1 \\cup \\Sigma_2$ in finitely many points (in fact, in exactly two points).\n\\end{enumerate}\nThis means exactly that $A$ determines an $\\mathbb{A}^1$-family of 1-cycles of perversity $\\overline 1$, so that $\\underline \\alpha_0 \\sim_{\\overline 1} \\underline \\alpha_1$.  (Of course $\\underline \\alpha_0$ and $\\underline \\alpha_1$ are then weakly equivalent as 1-cycles of perversity $\\overline 1$.)\n\nWe consider $[\\underline  \\alpha_0], [ \\underline  \\alpha_1], [\\Sigma_1],$ and $[\\Sigma_2]$ as points of $\\check P$, the $\\bb{P}^2$ dual to $P$.  The lines $ \\underline \\alpha_0, \\underline  \\alpha_1, \\Sigma_2$ meet at $[0:0:1:1:0]$, and this point does not belong to $\\Sigma_1$.  Therefore $[\\underline  \\alpha_0], [ \\underline  \\alpha_1],$ and $[\\Sigma_2]$ lie on a line $\\ell \\hookrightarrow \\check P$, and $[\\Sigma_1] \\notin \\ell$.  Then the family $A$ is the family of lines corresponding to the canonical morphism $\\ell - [\\Sigma_2] \\to \\check P$.  Since this family avoids $[\\Sigma_1]$ and $[\\Sigma_2]$, all of the fibers $A_t$ meet $\\Sigma_1 \\cup \\Sigma_2$ exactly twice.  Furthermore, exactly one point $[L] \\in \\ell$ corresponds to a line containing $p$; this is $[\\Sigma_2]$, hence all of the fibers $A_t$ are disjoint from $p$.\n\nThe incidences are contained in the locus where $S \\neq 0$, so are captured by the affine situation.\n\\end{proof}\n\n\n\\begin{remark}\nIn this example, the family $A$ of 1-cycles cannot be extended to a $\\bb{P}^1$-family respecting the perversity condition $\\overline 1$.  However, by performing the blowups in a different order, one can find an example of the same essential phenomenon, and so that the $\\mathbb{A}^1$-family relating the 1-cycles extends to a $\\bb{P}^1$-family, all of whose fibers satisfy the perversity condition $\\overline 1$.  Namely, we first blow up $\\Sigma_2$ on $\\underline X$, then blow up the proper transform of $\\Sigma_3$.  The resulting variety $\\underline X'$ is singular exactly along ${\\Sigma'_1}$, the proper transform of $\\Sigma_1$.  There are no incidences $\\underline D' \\cap \\underline \\alpha_i'$ in the exceptional divisors over $\\Sigma_2 \\cup \\Sigma_3$, so the incidences are captured by the affine situation.  The variety $\\underline X'$ is resolved by blowing up $\\Sigma_1$, and $B_{\\Sigma'_1} (\\underline X')$ has one-dimensional fibers over $\\Sigma'_1$, all of which are irreducible except one (the same which appears in the affine example).\n\nNote $\\Sigma_2 \\hookrightarrow P$ is already Cartier, so the proper transform of $P$ via the blowup along $\\Sigma_2$ is isomorphic to $P$.  Since $\\Sigma_3$ meets $P$ in a single point, the transform $P'$ of $P$ on $\\underline X'$ is isomorphic to $\\bb{P}^2$ blown up at a single point.  \n\nThe blown up point is not contained in any of the lines $\\Sigma_1', \\underline \\alpha'_0, \\underline \\alpha'_1$.  The rational function $P \\dashrightarrow \\bb{P}^1$ relating $\\underline \\alpha_0$ to $\\underline \\alpha_1$ may be considered as a rational function $P' \\dashrightarrow \\bb{P}^1$ relating $\\underline \\alpha'_0$ to $\\underline \\alpha'_1$; since $\\Sigma'_1$ does not occur as a fiber of this map, we may think of this function as a $\\bb{P}^1$-family of 1-cycles of perversity $\\overline 1$ for the stratification ${(\\underline X')}^3 = \\emptyset \\hookrightarrow {(\\underline X')}^2 = {(\\underline X')}^1 = \\Sigma'_1 \\hookrightarrow \\underline X'$.\n\\end{remark}\n\n\\section{Further questions} \\label{sec:further}\n\n\\textbf{Independence of resolution.}  Given two resolutions $\\pi_1 :X_1 \\to X \\ , \\ \\pi_2 : X_2 \\to X$ of the variety $X$, we have defined two stratifications $S_1, S_2$ of $X$, and (in certain situations) products $\\bullet_i : A_{r, \\overline p}(X, S_i) \\otimes A_{s, \\overline q}(X, S_i) \\to A_{r+s-d}(X)$ (for $i=1,2$) by pushing forward the intersection formed on the resolution.  (We make explicit the dependence of the group $A_{r, \\overline p}(X)$ on the stratification, and for simplicity we drop the superscript $w$ and the coefficients.)  For any stratification $S$ that refines both $S_1$ and $S_2$, we have canonical morphisms $C^i_{r, \\overline p}: A_{r,  \\overline p}(X, S) \\to A_{r, \\overline p}(X, S_i)$ (for $i=1,2$), and thus it makes sense to ask whether $\\bullet_1 (C^1_{r, \\overline p} \\otimes C^1_{s, \\overline q})$ and $\\bullet_2 (C^2_{r, \\overline p} \\otimes C^2_{s, \\overline q} )$ coincide as morphisms $A_{r,  \\overline p}(X, S) \\otimes A_{s, \\overline q}(X, S) \\to A_{r+s-d}(X)$.  We assume we are in one of the situations in which the resolution is known to induce a well-defined product.\n\nThe simplest case is when $\\pi_2$ is obtained from $\\pi_1$ by a sequence of blowups of $X_1$ along smooth centers.  In this case we have a canonical morphism \n$C_{r, \\overline p} : A_{r,  \\overline p}(X, S_2) \\to A_{r, \\overline p}(X, S_1)$ (i.e., the stratification induced by $\\pi_2$ refines the one induced by $\\pi_1$).  \n\n\\begin{proposition} With the notation and hypotheses as above, suppose $f : X_2 \\to X_1$ is a composition of blowups along smooth centers, and set $\\pi_2 = \\pi_1 \\circ f : X_2 \\to X$.  Then we have $\\bullet_2 = \\bullet_1 (C_{r, \\overline p} \\otimes C_{s, \\overline q}) : A_{r, \\overline p}(X,S_2) \\otimes A_{s, \\overline q}(X, S_2) \\to A_{r+s-d}(X)$. \\end{proposition}\n\n\\begin{proof}  Let $(-)_i$ denote the proper transform of a cycle on $X$ via $\\pi_i : X_i \\to X$.  We have $f^* (\\alpha_1) = \\alpha_2 + e_\\alpha$ and $f^* (\\beta_1) = \\beta_2 + e_\\beta$.  Since $\\alpha_1 \\cdot \\beta_1 = f_* (f^* (\\alpha_1) \\cdot f^* (\\beta_1))$, it follows that\n$${(\\pi_1)}_* (\\alpha_1 \\cdot \\beta_1) = {(\\pi_2)}_* (\\alpha_2 \\cdot \\beta_2) + {(\\pi_2)}_*(\\alpha_2 \\cdot e_\\beta + e_\\alpha \\cdot \\beta_2 + e_\\alpha \\cdot e_\\beta).$$\nThe cycles $e_\\alpha$ and $e_\\beta$ may be thought of as error terms which first appear on one of the blowups occurring in the morphism $f$.  Therefore our arguments apply to show the vanishing of the cycle class of ${(\\pi_2)}_*(\\alpha_2 \\cdot e_\\beta + e_\\alpha \\cdot \\beta_2 + e_\\alpha \\cdot e_\\beta)$, and therefore we obtain the equality ${(\\pi_1)}_* (\\alpha_1 \\cdot \\beta_1) = {(\\pi_2)}_* (\\alpha_2 \\cdot \\beta_2).$\n\\end{proof}\n\nIf $k$ is an algebraically closed field of characteristic zero and $X$ is complete, then two resolutions of $X$ are related by a sequence of smooth blowups and blowdowns by the weak factorization theorem of Abramovich-Karu-Matsuki-W{\\l}odarczyk \\cite[Thm.~0.1.1]{AKMW}.  Since the intermediate varieties do not necessarily admit morphisms to $X$, it is not clear how to obtain a comparison of the products formed via two resolutions.  If the strong factorization conjecture \\cite[Conj~0.2.1]{AKMW} holds, however, then there is variety $Y$ admitting morphisms $f_i : Y \\to X_i$ which are compositions of blowups along smooth centers, and such that $\\pi_1 \\circ f_1 = \\pi_2 \\circ f_2 : Y \\to X$.  In this case we could conclude that the products defined using $X_1$ and $X_2$ agree upon restriction to $A_{r, \\overline p}(X, S_Y) \\otimes A_{s, \\overline q}(X, S_Y)$, i.e., the group formed using the stratification induced by the resolution $Y \\to X$.\n\n\n\n\n\\medskip\n\\textbf{Comparison with Goresky-MacPherson product.} It would be interesting to know whether our intersection product (when it is defined) agrees with the intersection pairing defined by Goresky-MacPherson via the cycle class mapping.  The difficulty in making the comparison is that there is no obvious way to take the proper transform of a topological cycle.  A cycle on $X$ gives rise to a canonical cycle on $\\widetilde X$ relative to $\\widetilde E$, and using this one can show the products agree after composing with the canonical map $H_*(X) \\to H_*(X, X^1)$.  For pairs of cycles with supports intersecting properly in each stratum, or more generally any pair which is weakly $(\\sim_{\\overline p}, \\sim_{\\overline q})$-equivalent to such a pair, our intersection product agrees with that of Goresky-MacPherson since in this case both may be described as taking the closure of an intersection product formed on the smooth locus.\n\n\\medskip\n\\textbf{Refinements for small perversities.} If $\\overline p + \\overline q < \\overline t$, is there a refinement\n$$A_{r, \\overline p}(X) \\otimes A_{s, \\overline q}(X) \\to A_{r+s - d, \\overline p + \\overline q}(X) \\ \\ ?$$\nThis seems difficult to achieve using resolutions.  For example, suppose $\\alpha \\in A_{r, \\overline 0}(X)$ and $D \\in A_{d -1, \\overline 0}(X)$.  The fibers of $\\widetilde \\alpha \\cap \\pi^{-1}(X^i) \\to \\alpha \\cap X^i$ are typically $(i-1)$-dimensional (the source is typically $(r-1)$-dimensional, and the target is typically $(r-i)$-dimensional).  To find a representative of $\\pi_* (\\widetilde \\alpha \\cdot \\widetilde D)$ in $A_{r-1, \\overline 0}(X)$, one would seek a divisor $\\widetilde D_1 \\sim \\widetilde D \\hookrightarrow \\widetilde X$ such that the image of $\\widetilde \\alpha \\cap \\pi^{-1}(X^i) \\cap \\widetilde D_1 \\to \\alpha \\cap X^i$ is not dense.  This means $\\widetilde D_1$ misses most fibers of the morphism $\\widetilde \\alpha \\cap \\pi^{-1}(X^i) \\to \\alpha \\cap X^i$.  One might try the following technique for moving divisors, at least in the quasi-projective case: find some effective divisor $D$ (with better incidence properties) such that $\\widetilde D + D$ is ample, then use $\\widetilde D_1 = H-D$ for some $H \\in |\\widetilde D + D|$.\nUnfortunately, the divisor $H$, being ample, will meet \\textit{every} fiber of the morphism $\\widetilde \\alpha \\cap \\pi^{-1}(X^i) \\to \\alpha \\cap X^i$ if $i-1 \\geq 1$.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\BB^n=\\{z\\in\\CC^n\\ |\\ |z|<1\\}$ be the unit ball in $\\CC^n$ and\ndenote by $Rat(\\BB^n, \\BB^N)$ the set of all proper holomorphic\nrational maps from  $\\BB^n$ to  $\\BB^N$. We say that $f, g\\in\nRat(\\BB^n, \\BB^N)$ are {\\it holomorphically equivalent} (or {\\it\nequivalent}, for short) if there are $\\sigma\\in Aut(\\BB^n)$ and\n$\\tau\\in Aut(\\BB^N)$ such that $f=\\tau\\circ g\\circ \\sigma$. By a\nwell-known result of Cima-Suffridge \\cite{CS}, $F$ extends\nholomorphically across the boundary $\\p {\\BB}^n$.\n\nFor the equal dimensional case $N=n$, Alexander \\cite{A77} proved\nthat $Rat(\\BB^n, \\BB^n)$ must be automorphisms for $n>1$.\nSubsequently, much effort has been paid to the classification of\n$Rat(\\BB^n, \\BB^N)$ with $N>n$. When $N \/ n$ is not too large, it\nturns out that the maps are equivalent to relatively simple ones. In\nfact, the classification problem had been done for $N\\leq 3n-3$ and\nthe maps turn out to be all monomial maps. The systematic\ninvestigations on the precise classification of $Rat(\\BB^n, \\BB^N)$\ncan be found in the work of\n\\cite{F82,Hu99,HJ01,Ha05,HJX06,CJY18,JY18}, etc. In \\cite{FHJZ10},\nFaran-Huang-Ji-Zhang constructed a family of  maps in $Rat(\\BB^n,\n\\BB^{3n-2})$, which cannot be equivalent to any polynomial maps.\nThis indicates\nthat the maps could be quite\ncomplicated when $N\\geq 3n-2$.\n\n\\medspace\n\nTo study maps in $Rat(\\BB^n, \\BB^N)$ , there are two geometric\nproblems which are of fundamental importance. The first one is the\nD'Angelo conjecture.\nFor any rational holomorphic map $H=\\frac{(P_1, ..., P_m)}{Q}$ on\n$\\CC^n$ where $P_j, Q$ are holomorphic polynomials with $(P_1, ...,\nP_m, Q)=1$, the {\\it degree} of $H$ is defined, as in algebraic\ngeometry,  to be $deg(H):=\\max\\{\\deg(P_j), deg(Q),  1\\le j\\le m\\}$.\nThe D'Angelo conjecture is as follows: For any $F\\in Rat(\\BB^n,\n\\BB^N)$, it should have\n\\[\ndeg(F) \\le \\begin{cases}\n2N-3,\\ & if\\ n=2,\\\\\n\\frac{N-1}{n-1},\\ & if\\ n\\ge 3.\n\\end{cases}\n\\]\nThere are several partial results supporting this conjecture. The\nconjecture is true for all monomial maps, as demonstrated   by\nD'Angelo-Kos-Riehl \\cite{DKR03} for the case  $n=2$ and\nby  Lebl-Peter \\cite{LP12} for the case  $n \\ge 3$. If $F$ is a\nrational map with geometric rank one, this conjecture was proved in\n\\cite[corollary 1.3]{HJX06}. If the conjecture is proved, it would\nbe sharp due to the known examples. Also, it is proved that\n$deg(F)\\le \\frac{N(N-1)}{2(2n-3)}$ holds for any $F\\in Rat(\\BB^n,\n\\BB^N)$ with $n=2$ in \\cite{Me06} and with $n\\ge 2$ in \\cite{DL09}.\n\n\\medskip\n\nAnother geometric problem is the gap conjecture.\nFor any integer $k$ with $1\\le k\\le K(n)$ where $K(n):=\\max\\{t\\in\n\\ZZ^+\\ |\\ \\frac{t(t+1)}{2}<n\\}$, we recall the {\\it gap interval}\n(cf. \\cite{HJY09})\n\\begin{equation}\n{\\cal I}_k:=\\bigg(k n,\\ (k+1)n - \\frac{k(k+1)}{2} \\bigg).\n\\end{equation}\nThe  {\\it gap conjecture}, first raised in \\cite{HJY09}, is stated\nas follows: Any proper holomorphic rational map $F\\in  Rat(\\BB^n,\n\\BB^N)$ where $n\\ge 3$ is equivalent to a map of the form $(G, 0')$\nwhere $G\\in Rat(\\BB^n, \\BB^{N'})$ where $N'<N$ if and only if $N\\in\n{\\cal I}_k$ for some $1\\le k\\le K(n)$. The gap conjecture  for the\ncases of $k=1,2,3$ have been proved \\cite{Hu99,HJ01,HJY14}.\nRecently, P. Ebenfelt \\cite{Eb16} proposed a SOS conjecture (i.e.,\nthe Sums of Squares of Polynomial conjecture)  and proved that if\nthe SOS conjecture is true, then it implies the gap conjecture.\n\n\\medskip\n\nThe first gap interval is ${\\cal I}_1=(n, 2n-1)$. Huang \\cite{Hu99}\nshowed that $deg(F)=1$ if $N\\in {\\cal I}_1$.  When $N=2n-1$, it was\nproved by Faran \\cite{F82} that $deg(F)\\le 3$ for $n=2$ and by\nHuang-Ji \\cite{HJ01}  that $deg(F)\\le 2$ for $n>2$. The second gap\ninterval is ${\\cal I}_2=(2n, 3n-3)$. When $N \\le 3n-3$ (and $n\\ge\n4$), we know from \\cite{AHJY15} that $deg(F)\\le 2$. These results\nconfirm the D'Angelo conjecture for the first and the second gap\nintervals.\n\n\\medskip\n\nThe third gap interval is ${\\cal I}_3=(3n, 4n-6)$. If D'Angelo\nconjecture is true, we would have $deg(F)\\le 3$ for any $F\\in\nRat(\\BB^n, \\BB^{4n-6})$ because $deg(F)\\le\n\\frac{4n-6-1}{n-1}=4-\\frac{3}{n-1}$. This is confirmed by our main\nresult of this paper as follows.\n\n\\medskip\n\n\\begin{thm}\\label{mainthm}\n    If $F\\in Rat(\\BB^n, \\BB^{4n-6})$ with $n \\ge 7$, then $deg(F)\\le 3$.\n\\end{thm}\n\nThe rest of the paper is organized as follows. In Section 2, we\nintroduced some known properties for Rat$(\\HH_n,\\HH_N)$, especially\nfor maps of geometric rank $2$. Section 3 was devoted to the proof\nof our main theorem assuming Proposition \\ref{propdeg}. In Sections\n4-7, we gave a detailed proof of Proposition \\ref{propdeg} according\nto four different cases.\n\n\n\\medspace\n\n\n\\medskip\n\n\\section{Preliminaries}\n\n\n\nLet $\\HH_n=\\{(z, w)\\in\\CC^{n-1}\\times \\CC\\ |\\ \\text{Im}(w)>|z|^2\\}$\nbe the Siegel upper half space and denote by $Rat(\\HH_n, \\HH_N)$ the\nset of all proper holomorphic rational maps from $\\HH_n$ to $\\HH_N$.\nBy the Cayley transform, we can identify $\\BB^n$ with  $\\HH_n$ and\nidentify $Rat(\\BB^n,\\BB^N)$ with $Rat(\\HH_n, \\HH_N)$. In what\nfollows, we will prove Theorem \\ref{mainthm} through the properties\nof $Rat(\\HH_n, \\HH_N)$.\n\n\n\nLet $F=(f,\\phi,g)=(\\widetilde{f}, g)= (f_1,\\cdots$, $f_{n-1}$,\n$\\phi_1,\\cdots$, $\\phi_{N-n},g)\\in Rat(\\HH_n, \\HH_{N})$. For each\n$p\\in \\p\\HH_n$, define $\\sigma^0_p\\in \\hbox{Aut}(\\HH_n)$ and\n$\\tau^F_p\\in\\hbox{Aut}(\\HH_N)$ as follows:\n\\begin{equation*}\\begin{split}\n&\\sigma^0_p(z,w)=(z+z_0, w+w_0+2i \\langle z,\\overline{z_0}\n\\rangle),\\\\\n&\\tau^F_p(z^*,w^*)=(z^*-\\widetilde{f}(z_0,w_0),w^*-\\overline{g(z_0,w_0)}-\n2i \\langle z^*,\\overline{\\widetilde{f}(z_0,w_0)} \\rangle).\n\\end{split}\\end{equation*}\n Then\n$F$ is equivalent to $F_p:=\\tau^F_p\\circ F\\circ\n\\sigma^0_p=(f_p,\\phi_p,g_p)$ and  $F_p(0)=0$. In \\cite{Hu99}, Huang\nconstructed an automorphism $\\tau^{**}_p\\in\n  {Aut}_0({\\HH}_N)$ such that\n$F_{p}^{**}:=\\tau^{**}_p\\circ F_p$ satisfies the following\nnormalization:\n\\[\nf^{**}_{p}=z+{\\frac{i}{ 2}}a^{**(1)}_{p}(z)w+o_{wt}(3),\\ \\phi_p^{**}\n={\\phi_p^{**}}^{(2)}(z)+o_{wt}(2), \\ g^{**}_{p}=w+o_{wt}(4).\n\\]\n\nWrite\n$\\mathcal{A}(p):=-2i(\\frac{\\partial^2(f_p)^{\\ast\\ast}_l}{\\partial\nz_j\n    \\partial w}|_0)_{1\\leq j,l\\leq n-1}$. The {\\it geometric rank} of $F$ at\n$p$ is defined to be  the rank of the $(n-1)\\times (n-1)$ matrix\n$\\mathcal{A}(p)$, which is denoted by $Rk_F(p)$. Now we define the\n{\\it geometric rank} of $F$ to be $\\kappa_0(F)=max_{p\\in\n\\partial\\HH_n} Rk_F(p)$. \\medspace\n\nWhen a map in $ Rat({\\HH}_n,{\\HH}_N)$ is not of full rank (i.e.,\n$\\kappa_0\\le n-2$),  by the works of \\cite{Hu03} and \\cite{HJX06},\nit can further be normalized to the following form:\n\\begin{thm}\n    \\label{normalize **}\n    Suppose that $F\\in Rat({\\HH}_n,{\\HH}_N)$\n    has geometric rank  $1\\le\\kappa_0\\le n-2$ with $F(0)=0$. Then there are\n    $\\sigma\\in \\hbox{Aut}({\\HH}_n)$ and\n    $\\tau\\in \\hbox{Aut}({\\HH}_N)$ such that\n    $\\tau\\circ F\\circ \\sigma $ takes\n    the following form, which is still denoted by $F=(f,\\phi,g)$ for\n    convenience of notation:\n\n   \n    \\begin{equation}\n    \\left\\{\n    \\begin{array}{l}\n    f_l=\\sum_{j=1}^{\\kappa_0}z_jf_{lj}^*(z,w),\\ l\\le\\kappa_0,\\\\\n    f_j=z_j,\\  \\text{for} \\ \\kappa_0+1\\leq j\\leq n-1,\\\\\n    \\phi_{lk}=\\mu_{lk}z_lz_k+\\sum_{j=1}^{\\kappa_0}z_j\\phi^*_{lkj}\\ \\text{for\n    } \\ \\ (l,k)\\in {\\cal S}_0,\\\\\n    \\phi_{lk}=O_{wt}(3),\\ \\ (l,k)\\in {\\cal S}_1,\\\\\n    g=w;\\\\\n    f_{lj}^*(z,w)=\\delta_l^j+\\frac{i\\delta_{l}^j\\mu_l}{2}w+b_{lj}^{(1)}(z)w+O_{wt}(4),\\\\\n    \\phi^*_{lkj}(z,w)=O_{wt}(2),\\ \\  (l,k)\\in {\\cal S}_0,\\\\\n    \\phi_{lk}=\\sum_{j=1}^{\\kappa_0}z_j\\phi_{lkj}^*=O_{wt}(3)\\ \\ for\\\n    (l,k)\\in {\\cal S}_1.\n    \\end{array}\\right.\n    \\label{eqn:hao}\n    \\end{equation}\n    Here, for $1\\le \\kappa_0\\le n-2$, we write ${\\cal S} ={\\cal S}_0\\cup\n    {\\cal S}_1$, the index set  for all components of $\\phi$, where\n    ${\\cal S}_{0}=\\{(j,l): 1\\le j\\leq \\kappa_0, 1\\leq l\\leq n-1, j\\leq\n    l\\}$, $ {\\cal S}_1=\\Big\\{(j, l): j=\\kappa_0+1, \\kappa_0+1\\le l \\le \\kappa_0 +\n    N-n-\\frac{(2n-\\kappa_0-1)\\kappa_0}{2} \\Big\\}$, and\n    \\begin{equation}\n    \\label{mui and mujk} \\mu_{jl}=\\begin{cases}\\sqrt{\\mu_j+\\mu_l} &\\\n    for\\ j<l\\le\n    \\kappa_0; \\\\\n    \\sqrt{\\mu_j} & if \\ j\\le \\kappa_0 < l\\ or\\ if\\ j=l\\le \\kappa_0.\n    \\end{cases}\n    \\end{equation}\n\\end{thm}\n\nWrite $\\phi=(\\Phi_0, \\Phi_1)$ where $\\Phi_0=(\\phi_{lk})_{(l,k)\\in\n{\\cal S}_0}$, $\\Phi_1=(\\phi_{lj})_{(l,k)\\in{\\cal S}_1}$. Define\n$\\Phi_0^{(1,1)}(z)=\\sum^{\\kappa_0}_{j=1} e_j z_j$,\n$\\Phi_1^{(1,1)}(z)=\\sum^{\\kappa_0}_{j=1} \\hat{e_k}z_j$, $e^*_j=(e_j,\n\\hat e_j)$, and $\\xi_j(z)=\\ov{e_j}\\cdot \\Phi^{(2,0)}_0(z)$. Here we\nuse notation $\\Phi^{(s,t)}(z)$ to denote a homogeneous polynomial of\ndegree $s$ in $z$, and\n\\begin{equation*}\n\\begin{split}\n&\\Phi^{(s,t)}(z, w)=\\Phi^{(s,t)}(z) w^t,\\\n\\Phi^{(s,t)}(z)=\\sum\\limits_{s_1+\\cdots+s_{n-1}=s}\n\\Phi^{(s_1I_1+\\cdots+s_{n-1}I_{n-1}+tI_n)}z_1^{s_1}\\cdots\nz_{n-1}^{s_{n-1}}.\n\\end{split}\n\\end{equation*}\n\n\n\n\n\nLet  $F$ be as in Theorem \\ref{normalize **}. By \\cite[Corollary\n3.4]{HJY14}, we have the following explicit expression for\n$\\Phi^{(3,0)}_1(z)$.\n\\begin{lem}\n    \\label{phi30} Let $\\kappa_0\\ge 2$ and $(\\kappa_0+1)n-\\kappa_0\n    \\le N\\le (\\kappa_0+2)n - \\kappa_0^2 -2$. Then, after applying\n    a unitary transformation to the $\\Phi_1$-components,\n    $F$ is equivalent to another map such that the new map\n    (still denoted as $F$) has the property\n    \\begin{equation}\n    \\label{Phi(3,0)1}\n    \\Phi^{(3,0)}_1(z)=\\bigg(\\frac{2}{\\sqrt{\\mu_j+\\mu_l}}\n    \\bigg[\\sqrt{\\frac{\\mu_j}{\\mu_l}}z_j \\xi_l -\n    \\sqrt{\\frac{\\mu_l}{\\mu_j}} z_l \\xi_j\\bigg],\\ \\ 0'  \\bigg)_{1\\le\n        j<l\\le \\kappa_0}.\n    \\end{equation}\n   \n   \n\\end{lem}\nWe call a map satisfying Theorem \\ref{normalize **} and\n(\\ref{Phi(3,0)1}) a {\\it normalized map}, and denote it by\n$F^{***}$.\n\n\\medspace\n\nFrom the above lemma and \\cite[(3.7)]{HJY14}, we obtain\n\n\\begin{equation}\n\\label{phi (3,0)} \\phi_{jk}^{(3,0)}(z)=\\begin{cases}\n\\frac{2i}{\\mu_{jk}}[z_j \\xi_k(z) + z_k \\xi_j(z)],\\ \\ if\\ j<k\\le \\kappa_0,\\\\\n\\frac{2i}{\\mu_{jk}} z_l \\xi_j(z),\\ \\ if\\ j=k,\\ or\\ j\\le \\kappa_0 <k,\n\\end{cases}\n\\end{equation}\nwhere $\\xi=(\\xi_1, ..., \\xi_{\\kappa_0})=\\ov{(e_1, ...,\ne_{\\kappa_0})} \\cdot \\Phi_0^{(2,0)}$.\n\\medskip\n\nNext, we show more properties for $F\\in Rat({\\HH}_n,{\\HH}_{4n-6})$\nwith $n\\geq 7$ and geometric rank $\\kappa_0=2$.\n\n\nFrom (\\ref{Phi(3,0)1}), we have\n\\begin{equation}\n\\label{phi 1}\n\\begin{cases}\n\\phi_{33}^{(3I_1)} = \\frac{2\\mu_1}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}}\n\\ov{e_{2,11}},\\ \\ \\phi_{33}^{(3I_2)} = - \\frac{2\n\\mu_2}{\\sqrt{\\mu_1(\\mu_1+\\mu_2)}} \\ov{e_{1,22}}, \\ \\\n\\phi_{33}^{(3I_k)} = 0,\\\\\n\\phi_{33}^{(2I_1+I_2)} = \\frac{2\\sqrt{\\mu_1}}{\\sqrt{\\mu_2}}\n\\ov{e_{2,12}} - \\frac{2\\sqrt{\\mu_2}}{\\sqrt{\\mu_1+\\mu_2}}\n\\ov{e_{1,11}}, \\ \\\n\\phi_{33}^{(2I_1+I_j)}=\\frac{2 \\mu_1}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}} \\ov{e_{2,1j}},\\\\\n\\phi_{33}^{(I_1+I_2+I_j)} = \\frac{2\\sqrt{\\mu_1}}{\\sqrt{\\mu_1+\\mu_2}}\n\\ov{e_{2,2j}} - \\frac{2\\sqrt{\\mu_2}}{\\sqrt{\\mu_1+\\mu_2}} \\ov{e_{1,1j}},\\\\\n\\phi_{33}^{(I_1+2I_2)} = \\frac{2\\sqrt{\\mu_1}}{\\sqrt{\\mu_1+\\mu_2}}\n\\ov{e_{2,22}} - \\frac{2\\sqrt{\\mu_2}}{\\sqrt{\\mu_1}} \\ov{e_{1,12}}, \\\n\\\n\\phi_{33}^{(2I_2+I_j)} = - \\frac{2 \\mu_2}{\\sqrt{\\mu_1(\\mu_1+\\mu_2)}} \\ov{e_{1,2j}},\\\\\n\\phi_{33}^{(I_1+2I_k)} = \\phi_{33}^{(I_2+2I_k)} =\n\\phi_{33}^{(I_1+I_k + I_j)} = \\phi_{33}^{(I_2+I_k+I_j)} =\n\\phi_{33}^{(I_i+I_k+I_l)} = 0.\n\\end{cases}\n\\end{equation}\n\nFrom \\cite[(4.20) and (4.46)]{HJY14}, we know\n\\begin{equation}\n\\label{phi 2}\n\\begin{split}\n& \\phi_{11}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j}\ne_{j,11}\n= \\frac{-2i}{\\sqrt{\\mu_1}}z_1 f_1^{(1,2)}(z),\\\\\n&\\phi_{12}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j} e_{j,12}\n= \\frac{-2i}{\\sqrt{\\mu_1+\\mu_2}}(z_1 f_2^{(1,2)}(z)+z_2 f_1^{(1,2)}(z)),\\\\\n& \\phi_{22}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j}\ne_{j,22}\n= \\frac{-2i}{\\sqrt{\\mu_2}}z_2 f_2^{(1,2)}(z),\\\\\n& \\phi_{1\\alpha}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j}\ne_{j,1\\alpha}\n= \\frac{-2i}{\\sqrt{\\mu_1}}z_\\alpha f_1^{(1,2)}(z),\\\\\n& \\phi_{2\\alpha}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j}\ne_{j,2\\alpha}\n= \\frac{-2i}{\\sqrt{\\mu_2}}z_\\alpha f_2^{(1,2)}(z),\\\\\n& \\phi_{33}^{(2,1)}(z) - 2i \\sum^2_{j=1} \\frac{\\xi_j}{\\mu_j}\ne_{j,33} =\n\\frac{-2}{\\sqrt{\\mu_1+\\mu_2}}\\bigg(\\sqrt{\\frac{\\mu_1}{\\mu_2}} z_1\nf_2^{(1,2)}(z)\n-\\sqrt{\\frac{\\mu_2}{\\mu_1}} z_2 f_1^{(1,2)}(z)\\bigg).\\\\\n\\end{split}\n\\end{equation}\n\n\\medspace\n\nBy \\cite[(4.3)]{HJY14}, we have $\\frac{i}{2} \\mu_1\na_n^{(1)}(\\epsilon)+f_1^{(1,2)}(\\epsilon,0,...,0)=0,\n\\frac{i}{2}\\mu_2 b_n^{(1)}(\\epsilon)+f_2^{(1,2)}(\\epsilon,0,...,0)$\n$=0$, and $\\phi^{(1,2)}(\\epsilon,0,...,0)+e_1^* a_n^{(1)}(\\epsilon)\n+ e_2^* b_n^{(1)}(\\epsilon)=0$. Thus\n\\begin{equation}\n\\label{phi 3} \\phi^{(1,2)}(\\chi, 0)=-\\frac{2i}{\\mu_1}\nf_1^{(1,2)}(\\chi, 0) e_1^* - \\frac{2i}{\\mu_2} f_2^{(1,2)}(\\chi, 0)\ne_2^*.\n\\end{equation}\n\n\\medspace\n\n\nNext we recall the definitions and the properties of the degeneracy\nranks. These invariants are introduced by Lamel \\cite{La01} and\nEbenfelt-Huang-Zaitsev \\cite{EHZ04}.\n\nLet $F\\in Rat(\\HH_n,\\HH_N)$ and $p\\in \\p\\HH_n$. Write $L^j=\\frac{\\p\n}{\\p z_j} + 2i \\ov{z_j} \\frac{\\p}{\\p w}$. Then $\\{L^j\\}_{1\\le j\\le\nn-1}$ forms a basis of tangent vector fields of $(1,0)$ on\n$\\p\\HH_n$. Denote by $\\hat \\rho(Z, \\ov Z)$  the defining function of\nthe real hypersurface $\\p\\HH_N$, and denote by $\\hat \\rho_{\\ov\nZ}:=\\ov\\p \\hat \\rho$ the complex gradient of $\\rho$. Now we define\nan increasing sequence of linear subspaces $E_{k}(p) \\subset\n\\CC^{N}$ as follows:\n\\begin{equation}\nE_{k}(p)=span_{\\CC}\\{L^{\\alpha_1} ... L^{\\alpha_{n-1}} \\hat\\rho_{\\ov\nZ} \\circ F(p)\\ |\\ \\ |\\alpha_1|+...+|\\alpha_{n-1}|\\le k\\}\n\\end{equation}\nDefine $d_1(p):=0$ and $ \\label{dk(p)} d_k(p):= \\dim_\\CC \\\nE_k(p)\/E_1(p). $  By moving $p$ to a nearby point $p_0$ if\nnecessary, we may assume that all $d_l(p)$ are locally constant near\n$p_0$ and\n\\begin{equation}\n\\label{rank} d_2(p)<d_3(p)<...<d_{l_0}(p)=d_{l_0+1}(p)=....\n\\end{equation}\nfor some $l_0$ with $1\\le l_0\\le N-n+1$.  By [EHZ04], $l_0$  is\ncalled the {\\it degeneracy rank} of $F$, and  $d_{l_0}$ is called\nthe {\\it degeneracy dimension} of $F$. These definitions depend on\nthe open subset $U$.  {\\it degeneracy rank} $l_0$ of $F$ is defined\nto be the\n minimization of $l_0$ among all these open sets. Then we have the\n following degree estimates when both $\\kappa_0$ and $l_0$ are\n small.\n\n\n\n\\begin{thm} \\label{nondegenerate rank}\n    (\\cite[Theorem 1.4]{CJY18})\\ \\ \\\n    Let $F\\in Rat(\\BB^n, \\BB^N)$ with $5\\le n\\le N$. Suppose that  $\\kappa_0\\le 2$ and $l_0\\le 2$. Then $deg(F)\\le 2$.\n\\end{thm}\n\n\n\nFinally, we finish this section by recalling a lemma of \\cite{HJ01},\nwhich gives a reduction for the degree estimates of a rational\nholomorphic map.\n\n For any point $q=(\\w q, q_n)\\in \\CC^n$, the Segre family of\n$\\p\\HH_n$ is a family given by $Q_q:=\\{(\\zeta, \\tau)\\in\n\\CC^{n-1}\\times \\CC\\ |\\ \\tau=\\ov{q_n}+ 2i \\langle \\zeta, \\w\nq\\rangle) \\}$. Let $F:\\CC^n\\to \\CC^N$ be a rational holomorphic map.\nThen the restriction $F|_{Q_q}(\\zeta)= F(\\zeta, \\ov{q_n}+2i \\langle\n\\zeta, \\w q\\rangle)$ is a rational holomorphic map in $\\zeta\\in\n\\CC^{n-1}$.\n\n\n\n\\begin{lem}\n\\label{degree} (\\cite[lemmas 5.3-5.4]{HJ01}) Let $F=\\frac{(P_1, ...,\nP_N)}{Q}$ be a rational holomorphic map from $\\CC^n$ into $\\CC^N$\nwhere $P_j$ and $Q$ are as above. Suppose that there exists a fixed\npositive integer $k$ such that one of the following two conditions\nis satisfied:\n\\begin{enumerate}\n\\item $deg(F|_{Q_q})\\le k$ for all $q\\in \\p\\HH_n$.\n\\item $deg(F^{**}_q|_{Q_0})\\le k$ for all $q\\in \\p\\HH_n$.\n\\end{enumerate}\nThen $deg(F)\\le k$.\n\\end{lem}\n\n\\section{Proof of Theorem \\ref{mainthm}}\n\nThis section is devoted to the proof of Theorem \\ref{mainthm},\nassuming Proposition \\ref{propdeg}, whose proof will be given in the\nend of this section and in $\\S 4-7$.\n\nWe start with  some notations. For any $F\\in Rat(\\HH_n, \\HH_{4n-6})$\nand $q\\in \\p \\HH_n$, the map associates to a normalized map\n$F_q^{***}$. We write $F_q^{***}=(f^{***}_q, \\phi^{***}_q,\ng^{***}_q)$ and $\n\\phi_{q}^{***}=\\big((\\Phi_{q}^{***})_0,(\\Phi_{q}^{***})_1\\big) $,\nwhich are decomposed similar to those of $F$ in Theorem\n\\ref{normalize\n**}.\n\n\\begin{lem}\n\\label{lemma N=3n-2} Let $F\\in Rat(\\HH_n, \\HH_{4n-6})$ be a\nnormalized map with $\\kappa_0=2$ and $n\\ge 7$. Then\n\\begin{equation}\n\\label{lemma 2.5 1} D^\\alpha F(0)\\in span_{|\\beta|\\le 3} \\{D^\\beta\nF(0)\\}\n\\end{equation}\nfor any index $\\alpha$,  where $D^\\alpha=\\frac{\\p^{|\\alpha|}}{\\p\nz_1^{\\alpha_1}\\cdots \\p z_n^{\\alpha_n}}$ is the standard\ndifferential operator. If, in addition,\n$\\big(\\Phi^{***}_{q}\\big)_1^{(3,0)}(z)\\equiv 0$ for any\n $q$  in\nan open neighborhood of $0$ in $\\p\\HH_n$, then for any index\n$\\alpha$,\n\\begin{equation}\n\\label{lemma 2.5 2} D^\\alpha F(0)\\in span_{|\\beta|\\le 2} \\{D^\\beta\nF(0)\\}.\n\\end{equation}\n\\end{lem}\n\n\\medspace\n\n\\begin{proof} By \\cite[Theorem 4.1 and (5.3)]{HJY14}, we have\n$span_{|\\beta|\\le 4}\\{ {\\cal L}^\\beta F|_0\\} \\le span_{|\\alpha|\\le\n3}\\{ {\\cal L}^\\alpha F|_0\\}$. By \\cite[claim (5.5)]{HJY14}, for any\n$p\\in\\p\\HH_n$, the associated map $F_p$ satisfies\n\\[\nspan_{|\\beta|\\le 4}\\{ {\\cal L}^\\beta F_p|_0\\} \\le span_{|\\alpha|\\le\n3}\\{ {\\cal L}^\\alpha F_p|_0\\}.\n\\]\nSince ${\\cal L}^\\alpha (F_p)|_0={\\cal L}^\\alpha F(p)$, it implies\n$span_{|\\beta|\\le 4}\\{ {\\cal L}^\\beta F|_p\\} \\le span_{|\\alpha|\\le\n3}\\{ {\\cal L}^\\alpha F|_p\\}$. Then  by similar argument as that in\n\\cite[p. 139]{HJY14}, through  applying $\\ov{\\cal L}_j, {\\cal L}_k$\nto $F$, we obtain (\\ref{lemma 2.5 1}). If, in addition,\n{$(\\Phi^{***}_q)_1^{(3,0)}(z)\\equiv 0$ for any\n $q$  in\nan open neighborhood of $0$ in $\\p\\HH_n$}, by the last paragraph of\n\\cite[p. 139]{HJY14}, we get (\\ref{lemma 2.5 2}).\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{mainthm}]\n\nAs mentioned in $\\S 2$, by the Cayley transform, we can\n identify $Rat(\\BB^n,\\BB^N)$ with $Rat(\\HH_n, \\HH_N)$. It suffices for us to show for\n  any $F\\in Rat(\\HH_n, \\HH_{4n-6})$ with $n\\geq 7$,\n  we have deg$(F)\\leq 3$.\n\n  By \\cite[Theorem 1.1]{Hu03}, we have $N \\ge n\n  +\\frac{(2n-\\kappa_0-1)\\kappa_0}{2}$. Hence for $N=4n-6$, its\n geometric rank $\\kappa_0\\le 3$. By \\cite{Hu99}, we can suppose $1\\leq \\kappa_0\\le 3$, otherwise the map must be linear fractional.\n  If  $F\\in Rat(\\HH_n,\n\\HH_{4n-6})$ with $n \\ge 3$ and $\\kappa_0=1$, by \\cite[Corollary\n1.3]{HJX06}, we have $deg(F)\\le 3$.  If $F\\in Rat(\\HH_n,\n\\HH_{4n-6})$ with $n \\ge 5$ and $\\kappa_0=3$, by  \\cite[ Theorem\n1.1]{JY18}, we have $deg(F)\\le 2$. Therefore, we assume $\\kappa_0=2$\nfor $F$ in the rest proof of Theorem \\ref{mainthm}.\n\n\n\n\n\nLet $F\\in Rat(\\HH_n, \\HH_{4n-6})$ with $n \\ge 7$ and $\\kappa_0=2$.\nFor any $q\\in \\p\\HH_n$, we study the associated map $(F_{q})^{***}$.\nThere are two cases to consider.\n\n\\medspace\n\n{\\bf Case I:} $F^{***}_{q}$ satisfies\n$\\big(\\Phi_{q}^{***}\\big)_1^{(3,0)}(z)\\equiv 0$ for any $q$ in a\nneighborhood of $0$ in $\\p\\HH_n$. Recall that $F^{***}_{q}$ is a\nnormalized map with geometric rank $2$. Hence by (\\ref{lemma 2.5\n2}), it implies that the degeneracy rank $l_0\\le 2$. Then we apply\nTheorem \\ref{nondegenerate rank} to conclude $deg(F)\\le 2$.\n\n\\medspace\n\n{\\bf Case 2:} $F^{***}_{q_0}$ satisfies that\n$\\big(\\Phi_{q_0}^{***}\\big)_1^{(3,0)}(z)\\not\\equiv 0$  for some\n$q_0\\in \\p\\HH_n$. As in Case I,  $F^{***}_{q_0}$ is a normalized map\nwith geometric rank $2$.  Now we need the following proposition.\n\n\\begin{prop}\\label{propdeg}\n  Let $F\\in Rat(\\HH_n,\\HH_{4n-6})\\ (n\\geq 7)$ be a normalized map, whose geometric rank\n  $\\kappa_0=2$. Suppose that $(\\Phi_1)^{(3,0)}(z)\\not\\equiv 0$.Then\n  \\begin{equation}\\label{reduction 0}\n deg(f(z, 0), \\phi(z, 0))\\le 3.\n\\end{equation}\n\\end{prop}\n\nThe proof of Proposition \\ref{propdeg} will be postponed to the next\nsection. Admitting this proposition temporarily, we continue the\nproof of Theorem \\ref{mainthm}.\n\nNotice that $(\\Phi_{q}^{***})_1^{(3,0)}(z)$ is smooth with respect\nto $q$. From $(\\Phi_{q_0}^{***})_1^{(3,0)}(z)\\not\\equiv 0$, we know\n$(\\Phi_{q}^{***})_1^{(3,0)}(z)\\not\\equiv 0$ for $q$ in a small\nneighborhood of $q_0$. Applying Proposition \\ref{propdeg} with $F$\nreplaced by $F_q^{***}$ for any $q\\in \\p\\HH_n$ in a neighborhood of\n$q_0$, we conclude  that deg$(F_q^{***}(z, 0))\\le 3$. Since maps in\nboth Aut$(\\p \\HH_n) $ and Aut$(\\p \\HH_N) $ are linear fractional, we\nknow deg$(F_q(z, 0))\\le 3$. Now we use Lemma \\ref{degree}  to\ncomplete the proof of Theorem \\ref{mainthm}.\n\\end{proof}\n\nNext we proceed to the proof of Proposition \\ref{propdeg}.\n\n\\begin{proof}[Proof of Proposition \\ref{propdeg}]\n\nBy \\cite[Lemma 2.3]{HJX06}, we know $f_l(z, 0)=z_l$. From Lemma\n\\ref{phi30}, $\\Phi_1=(\\phi_{j,k})_{(j,k)\\in {\\cal S}_1} =(\\phi_{33},\n\\phi_{34}, ..., \\phi_{3 (n-2)})$ satisfies $\\Phi^{(3)}_1(z,\n0)=(\\phi^{(3)}_{33}(z, 0), 0, ..., 0)$. Then we apply Lemma\n\\ref{lemma N=3n-2} to know\n\\begin{equation}\n\\label{reduction 2} \\phi_{34}(z, 0)\\equiv 0, \\cdots, \\phi_{3\n(n-2)}(z, 0)\\equiv 0.\n\\end{equation}\nHence  it suffices to prove\n\\begin{equation}\n\\label{reduction} deg\\bigg(deg(\\phi_{jk}(z, 0)), \\ deg(\\phi_{33}(z,\n0)\\bigg) \\le 3,\\ \\ \\ \\forall (j,k)\\in {\\cal S}_0.\n\\end{equation}\n\n\nBy our notation,\n$\\phi^{(3,0)}_{33}=(\\phi_{33}^{(j_1I_1+j_{n-1}I_{n-1})})_{j_1+\\cdots+j_{n-1}=3,j_1+j_2\\geq\n1 }$. From (\\ref{phi 1}) and (\\ref{reduction 2}),\n$(\\Phi_1)_{33}^{(3,0)}(z)\\not\\equiv 0$ if and only if\n\\begin{equation}\n\\label{cases} (\\phi_{33}^{(3I_1)}, \\phi_{33}^{(3I_2)},\n\\phi_{33}^{(2I_1+I_2)}, \\phi_{33}^{(I_1+2I_2)},\n\\phi_{33}^{(I_1+I_2+I_j)}, \\phi_{33}^{(2I_1+I_j)},\n\\phi_{33}^{(2I_2+I_j)})\\not=(0, 0, ...., 0).\n\\end{equation}\nThus it is suffices for us to consider the following cases:\n\\begin{equation}\\begin{split}\\label{ABCD}\n &\\text{Case A: } \\phi_{33}^{(3I_1)}\\not=0,\\ \\text{Case A}':\\\n \\phi_{33}^{(3I_2)}\\not=0,\\\\\n&\\text{Case B:}\\  \\phi_{33}^{(2I_1+I_2)}\\not=0, \\ \\text{Case B}': \\\n\\phi_{33}^{(I_1+2I_2)}\\not=0,\\\\\n&\\text{Case C:}\\ \\phi_{33}^{(I_1+I_2+I_j)}\\not=0,\\\\\n&\\text{Case D:} \\ \\phi_{33}^{(2I_1+I_j)}\\not=0, \\ \\text{Case D}':\\\n\\phi_{33}^{(2I_2+I_j)}\\not=0.\n\\end{split}\\end{equation}\n\n\n\n\nIn the rest of the paper, we will obtain an explicit expression for\n$\\phi_{33}(z, 0)$ and  $\\phi_{jk}(z, 0)$ with $(j,k)\\in {\\cal S}_0$,\nfrom which we obtain (\\ref{reduction}).\n\nWe start with the following Chern-Moser equation.\n\\[\n\\frac{g(z, w)-\\ov{g(z, w)}}{2i} = f(z, w)\\cdot \\ov{f(z, w)}+\\phi(z,\nw) \\cdot \\ov{\\phi(z, w)}, \\ \\ \\ \\forall\\ \\text{Im}(w)=|z|^2.\n\\]\nBy complexification, we write\n\\begin{equation}\n\\label{basic 1} \\frac{g(z, w) - \\ov{g(\\ov\\chi, \\ov\\eta)}}{2i} =\n\\sum^{n-1}_{l=1} f_l(z, w) \\ov{f_l(\\ov\\chi, \\ov\\eta)} +\n\\sum_{(s,t)\\in {\\cal S}} \\phi_{st}(z, w) \\ov{\\phi_{st}(\\ov\\chi,\n\\ov\\eta)},\\ \\ \\forall\\ \\frac{w-\\eta}{2i}=z\\cdot \\chi.\n\\end{equation}\nApplying ${\\cal L}_j:=\\frac{\\p }{\\p z_j} + 2i \\chi_j \\frac{\\p}{\\p\n    w}$ for $z=0$ and $w=\\eta=0$ to the both sides of the above\nidentity, we obtain\n\\begin{equation}\n\\label{basic 2} \\frac{{\\cal L}_j g(0, 0)}{2i} = \\sum^{n-1}_{l=1}\n{\\cal L}_j f_l(0, 0) \\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal\nS}_0} {\\cal L}_j\\phi_{st}(0, 0) \\ov{\\phi_{st}(\\ov\\chi, 0)},\n\\end{equation}\n\n\\begin{equation}\n\\label{basic 3} \\frac{{\\cal L}_j {\\cal L}_k g(0, 0)}{2i} =\n\\sum^{n-1}_{l=1} {\\cal\n    L}_j {\\cal L}_k f_l(0, 0) \\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal S}_0}\n{\\cal L}_j{\\cal L}_k \\phi_{st}(0, 0) \\ov{\\phi_{st}(\\ov\\chi, 0)}.\n\\end{equation}\nWe notice that the index set ${\\cal S}$ in (\\ref{basic 1}) is\nreplaced by ${\\cal S}_0$ in (\\ref{basic 2}) and (\\ref{basic 3})\nbecause $\\phi_{st}^{(2)}(0,0)=0$ for any $(s, t)\\in {\\cal S}_1$. The\nproof of Proposition \\ref{propdeg} will be completed in Sections\n$4-7$ according to the different cases in (\\ref{ABCD}).\n\\end{proof}\n\n\\section{Proof of Proposition \\ref{propdeg} for Case A}\n\nIn this section,  we'll proceed to complete the proof of Proposition\n\\ref{propdeg} for Case A.\n\n\\begin{proof}[Proof of Proposition \\ref{propdeg} for Case A]\nApplying ${\\cal L}_1^3$ to (\\ref{basic 1}), we obtain\n\\begin{equation}\n\\label{basic 4} \\frac{{\\cal L}_1^3  g(0, 0)}{2i} = \\sum^{n-1}_{l=1}\n{\\cal L}^3_1 f_l(0, 0) \\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal\nS}_0} {\\cal L}_1^3 \\phi_{st}(0, 0) \\ov{\\phi_{st}(\\ov\\chi, 0)} +\n{\\cal L}_1^3 \\phi_{33}(0, 0) \\ov{\\phi_{33}(\\ov\\chi, 0)}.\n\\end{equation}\nHere we have used (\\ref{reduction 2}). We can write these equations\nin terms of matrix,\n\n\\begin{equation}\\begin{split}\\label{matrixb}\nB\n\\begin{pmatrix}\n\\ov{f_1(\\ov\\chi, 0)}\\\\ \\ov{f_{2}(\\ov\\chi, 0)}\\\\\n\\ov{\\phi(\\ov\\chi, 0)}\n\\end{pmatrix} = \\begin{pmatrix}\n\\chi_1\\\\\n\\chi_{2}\\\\\n0\\\\\n\\vdots\\\\\n0\\\\\n\\end{pmatrix}\n\\end{split}\\end{equation}\nwhere  $B$ is a non-singular $(2n-1)\\times (2n-1)$ matrix evaluated\nat $(0,0)$:\n\n\\begin{equation}\\label{matrix}\nB:=\\begin{pmatrix} {\\cal L}_jf_{h} & {\\cal L}_j \\phi_{hl} &\n{\\cal L}_j \\phi_{h\\alpha} & {\\cal L}_j \\phi_{33}\\\\\n{\\cal L}_j{\\cal L}_kf_{h} &  {\\cal L}_j{\\cal L}_k \\phi_{hl}\n& {\\cal L}_j{\\cal L}_k \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_k \\phi_{33} \\\\\n{\\cal L}_j{\\cal L}_\\beta f_{h} &\n{\\cal L}_j{\\cal L}_\\beta \\phi_{hl} & {\\cal L}_j{\\cal L}_\\beta \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_\\beta \\phi_{33}\\\\\n{\\cal L}^3_1 f_{h}  & {\\cal L}^3_1 \\phi_{hl} &\n{\\cal L}^3_1 \\phi_{h\\alpha} & {\\cal L}^3_1 \\phi_{33} \\\\\n\\end{pmatrix}_{1\\leq j,h,l\\leq 2, {3\\leq \\alpha,\\beta\\leq n-1}}\n\\end{equation}\nIn the rest of this section, we'll use (\\ref{matrixb}) to solve\n$\\phi(z, 0)$.\n\n\\medskip\n\nBy our normalization (\\ref{eqn:hao}), we have\n\\begin{equation} \\label{L2 eq}\n\\begin{split}\n& {\\cal L}_1^2 f_1|_{(0,0)}=4i \\chi_1 f_1^{(I_1+I_n)} = - 2 \\mu_1 \\chi_1,\\ \\ {\\cal L}_1^2 f_2|_{(0,0)}=0, \\\\\n& {\\cal L}_1{\\cal L}_2 f_1|_{(0,0)} = 2i \\chi_2 f_1^{(I_1+I_n)} = -\n\\mu_1 \\chi_2,\\ \\\n{\\cal L}_1 {\\cal L}_2 f_2|_{(0,0)} = 2i \\chi_1 f_2^{(I_2+I_n)} = -\\mu_2 \\chi_1, \\\\\n& {\\cal L}_2^2 f_1|_{(0,0)} = 0,\\ \\ \\ {\\cal L}_2^2 f_2|_{(0,0)} = 4i \\chi_2 f_2^{(I_2+I_n)} = - 2 \\mu_2 \\chi_2, \\\\\n& {\\cal L}_1 {\\cal L}_\\alpha f_1|_{(0,0)} = 2i \\chi_\\alpha\nf_1^{(I_1+I_n)} = - \\mu_1 \\chi_\\alpha,\n\\ \\ {\\cal L}_1 {\\cal L}_\\alpha f_2|_{(0,0)} = 0,\\\\\n& {\\cal L}_2 {\\cal L}_\\alpha f_1|_{(0,0)} = 0,\\ {\\cal L}_2 {\\cal\nL}_\\alpha f_2|_{(0,0)} = 2i \\chi_\\alpha f_2^{(I_2+I_n)} = - \\mu_2\n\\chi_\\alpha.\n\\end{split}\n\\end{equation}\nNotice that $f_1^{(2I_1+I_n)}=-\\sqrt{\\mu_1}\\ov{e_{1,11}},\nf_2^{(2I_2+I_n)}= - \\sqrt{\\mu_1}\\ov{e_{2,11}}$. Recall ${\\cal\nL}_1^3=\\frac{\\p^3}{\\p z_1^3} + 6i \\chi_1 \\frac{\\p^3}{\\p z_1^2 \\p w}\n- 12\\chi_1^2 \\frac{\\p^3}{\\p z_1 \\p w^2} - 8i \\chi_1^3 \\frac{\\p^3}{\\p\nw^3}$.\nThen\n\\begin{equation}\n\\begin{split}\n{\\cal L}_1^3 f_1|_{(0,0)} & = 6i \\chi_1 \\cdot 2 f_1^{(2I_1+I_n)}\n-12\\chi_1^2 \\cdot 2 f_1^{(I_1+2I_n)}\\\\\n& = - 12i \\chi_1 \\cdot \\sqrt{\\mu_1} \\ov{e_{1,11}}-24\\chi_1^2 f_1^{(I_1+2I_n)}:=C_1, \\\\\n{\\cal L}_1^3 f_2|_{(0,0)} & =  6i \\chi_1 \\cdot 2 f_2^{(2I_1+I_n)}\n-12\\chi_1^2 \\cdot 2 f_2^{(I_1+2I_n)}\\\\\n& = - 12i \\chi_1 \\cdot \\sqrt{\\mu_1} \\ov{e_{2,11}} -24\\chi_1^2\nf_2^{(I_1+2I_n)}:\n= C_2. \\\\\n\\end{split}\n\\end{equation}\n\n\\medspace\n\nSet\n\\begin{equation}\nB_1:= \\begin{pmatrix}\n{\\cal L}_j{\\cal L}_k \\phi_{hl} & {\\cal L}_j{\\cal L}_k \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_k \\phi_{33} \\\\\n{\\cal L}_j{\\cal L}_\\beta \\phi_{hl} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{33}\\\\\n{\\cal L}^3_1 \\phi_{hl} & {\\cal L}^3_1 \\phi_{h\\alpha} &\n{\\cal L}^3_1 \\phi_{33} \\\\\n\\end{pmatrix}_{1\\leq j,h,l\\leq 2, {3\\leq \\alpha,\\beta\\leq n-1}} \\bigg|_{(0,0)}\n=\\begin{pmatrix}\nB_{1, jk}\\\\\nB_{1, j \\beta}\\\\\nB_{1, 33}\n\\end{pmatrix}.\n\\end{equation}\nThen (\\ref{matrixb}) takes the following form:\n\\begin{equation}\n\\label{new form} B_1 \\ov{\\phi}(\\chi, 0)^t=A_1,\n\\end{equation}\nwhere $A_1=(A_{1,11}, A_{1,12}, A_{1,22}, A_{1,1\\alpha},\nA_{1,2\\alpha}, A_{1,33})^t$:\n\\begin{equation}\n\\begin{split}\n& A_{1,11}=-{\\cal L}_1^2 f_1|_{(0,0)}\\chi_1 - {\\cal L}_1^2 f_2|_{(0,0)} \\chi_2 = 2 \\mu_1 \\chi_1^2,\\\\\n& A_{1,12}=-{\\cal L}_1{\\cal L}_2 f_1 |_{(0,0)}\\chi_1 - {\\cal\nL}_1{\\cal L}_2 f_2|_{(0,0)} \\chi_2\n= \\mu_1 \\chi_1 \\chi_2 + \\mu_2 \\chi_1\\chi_2,\\\\\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n& A_{1,22}=-{\\cal L}_2^2 f_1|_{(0,0)} \\chi_1 - {\\cal L}_2^2 f_2|_{(0,0)} \\chi_2 = 2\\mu_2 \\chi_2^2, \\\\\n& A_{1,1\\alpha} = - {\\cal L}_1{\\cal L}_\\alpha f_1|_{(0,0)} \\chi_1 - {\\cal L}_1{\\cal L}_\\alpha f_2|_{(0,0)} \\chi_2 = \\mu_1  \\chi_1 \\chi_\\alpha,\\\\\n& A_{1,2\\alpha} = -{\\cal L}_1{\\cal L}_\\alpha f_1|_{(0,0)} \\chi_1- {\\cal L}_1{\\cal L}_\\alpha f_2|_{(0,0)}\\chi_2 =\\mu_2 \\chi_2 \\chi_\\alpha,\\\\\n& A_{1,33}=-{\\cal L}_1^3 f_1|_{(0,0)} \\chi_1 - {\\cal L}_1^3\nf_2|_{(0,0)} \\chi_2 = -\\chi_1 C_1 - \\chi_2 C_2,\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\label{def of B1}\n\\begin{split}\nB_{1,11}&=(2\\mu_{11} + 4i \\chi_1 e_{1,11}, 4i \\chi_1 e_{1,12}, 4i\\chi_1 e_{1,22}, 4i\\chi_1 e_{1,1\\alpha}, 4i \\chi_1 e_{1,2\\alpha}, 4i \\chi_1 e_{1,33}),\\\\\nB_{1,12}&=(2 \\chi_1 e_{2,11} + 2i \\chi_2 e_{1,11}, \\mu_{12} + 2i \\chi_1 e_{2,12} + 2i \\chi_2 e_{1,12}, 2i\\chi_1 e_{2,22}+2i\\chi_2 e_{1,22}, \\\\\n&\\ \\ \\ \\ \\ \\ 2i\\chi_1 e_{2,1\\alpha}+2i\\chi_2 e_{1,1\\alpha}, 2i\n\\chi_1 e_{2,2\\alpha}\n+2i \\chi_2 e_{1,2\\alpha}, 2i \\chi_1 e_{2,33}+2i\\chi_2e_{1,33}),\\\\\nB_{1,22}&=(4 i \\chi_2 e_{2,11}, 4i \\chi_2 e_{2,12}, 2 \\mu_{22} + 4i\\chi_2 e_{2,22}, 4i\\chi_2 e_{2,1\\alpha}, 4i \\chi_2 e_{2,2\\alpha}, 4i \\chi_2 e_{2,33}),\\\\\nB_{1,1\\beta}&=(2i \\chi_\\beta e_{1,11}, 2i \\chi_\\beta e_{1,12},\n2i\\chi_\\beta e_{1,22},\n\\mu_{1\\alpha} \\delta^\\beta_\\alpha + 2i\\chi_\\beta e_{1,1\\alpha}, 2i \\chi_\\beta e_{1,2\\alpha}, 2i \\chi_\\beta e_{1,33}),\\\\\nB_{1,2\\beta}&=(2i \\chi_\\beta e_{2,11}, 2i \\chi_\\beta e_{2,12},\n2i\\chi_\\beta e_{2,22},\n2i\\chi_\\beta e_{2,1\\alpha}, \\mu_{2\\alpha} \\delta^\\beta_\\alpha + 2i \\chi_\\beta e_{2,2\\alpha}, 2i \\chi_\\beta e_{2,33}),\\\\\nB_{1,33} & = ({\\cal L}_1^3 \\phi_{11}, {\\cal L}_1^3 \\phi_{12}, {\\cal\nL}_1^3 \\phi_{22},\n{\\cal L}_1^3 \\phi_{1\\alpha},{\\cal L}_1^3 \\phi_{2\\alpha},  {\\cal L}_1^3 \\phi_{33}) |_{(0,0)}.\\\\\n\\end{split}\n\\end{equation}\nBy (\\ref{phi (3,0)})-(\\ref{phi 2}), $B_{1,33}$ can be calculated as\nfollows.\n\n\\begin{equation*}\n\\begin{split}\n&{\\cal L}_1^3 \\phi_{11}|_{(0,0)} = 6\\cdot 2 i \\ov{e_{1,11}} + 6i\n\\chi_1 \\cdot 2\\cdot 2i\n\\bigg(\\frac{\\sqrt{\\mu_1}}{\\mu_1} \\ov{e_{1,11}}e_{1,11} + \\frac{\\sqrt{\\mu_1}}{\\mu_2}\\ov{e_{2,11}}e_{2,11} \\\\\n&\\ \\ -\\frac{1}{\\sqrt{\\mu_1}} f_1^{(I_1+2I_n)}\\bigg)\n -12 \\chi_1^2 4i\\bigg(-\\frac{1}{\\mu_1}e_{1,11} f_1^{(I_1+2I_n)} - \\frac{1}{\\mu_2} e_{2,11} f_2^{(I_1+2I_n)}\\bigg)\\\\\n& = - \\frac{1}{\\sqrt{\\mu_1} \\chi_1} C_1 - 2i \\bigg(\\frac{e_{1,11}}{\\mu_1} C_1 + \\frac{e_{2,11}}{\\mu_2} C_2 \\bigg),\\\\\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n&{\\cal L}_1^3 \\phi_{12}|_{(0,0)} = 6\n\\frac{\\sqrt{\\mu_1}}{\\sqrt{\\mu_1+\\mu_2}} \\cdot 2 i \\ov{e_{2,11}} + 6i\n\\chi_1 \\cdot 2\\cdot 2i\n\\bigg(\\frac{\\sqrt{\\mu_1}}{\\mu_1} \\ov{e_{1,11}}e_{1,12} + \\frac{\\sqrt{\\mu_1}}{\\mu_2}\\ov{e_{2,11}}e_{2,12} \\\\\n&\\ \\ -\\frac{1}{\\sqrt{\\mu_1+\\mu_2}} f_2^{(I_1+2I_n)}\\bigg)\n-12 \\chi_1^2 4i\\bigg(-\\frac{1}{\\mu_1}e_{1,12} f_1^{(I_1+2I_n)} - \\frac{1}{\\mu_2} e_{2,12} f_2^{(I_1+2I_n)}\\bigg)\\\\\n& = - \\frac{1}{\\sqrt{\\mu_1+\\mu_2} \\chi_1} C_2 - 2i \\bigg(\\frac{e_{1,12}}{\\mu_1} C_1 + \\frac{e_{2,12}}{\\mu_2} C_2 \\bigg),\\\\\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n&{\\cal L}_1^3 \\phi_{22}|_{(0,0)} = 6i \\chi_1 \\cdot 2\\cdot 2i\n\\bigg(\\frac{\\sqrt{\\mu_1}}{\\mu_1} \\ov{e_{1,11}}e_{1,22} + \\frac{\\sqrt{\\mu_1}}{\\mu_2}\\ov{e_{2,11}}e_{2,22}\\bigg)\\\\\n&-12 \\chi_1^2 4i\\bigg(-\\frac{1}{\\mu_1}e_{1,22} f_1^{(I_1+2I_n)} -\n\\frac{1}{\\mu_2} e_{2,22} f_2^{(I_1+2I_n)}\\bigg)\n=  - 2i \\bigg(\\frac{e_{1,22}}{\\mu_1} C_1 + \\frac{e_{2,22}}{\\mu_2} C_2 \\bigg).\\\\\n\\end{split}\n\\end{equation*}\n\nFor $j=1$ or $2$, we have\n\n\\begin{equation*}\n\\begin{split}\n&{\\cal L}_1^3 \\phi_{j\\alpha}|_{(0,0)} = 6i \\chi_1 \\cdot 2\\cdot 2i\n\\bigg(\\frac{\\sqrt{\\mu_1}}{\\mu_1} \\ov{e_{1,11}}e_{1,j\\alpha} +\n\\frac{\\sqrt{\\mu_1}}{\\mu_2}\\ov{e_{2,11}}e_{2,j\\alpha}\n\\bigg)\\\\\n&\\ \\ \\ -12 \\chi_1^2 4i\\bigg(-\\frac{1}{\\mu_1}e_{1,j\\alpha}\nf_1^{(I_1+2I_n)} - \\frac{1}{\\mu_2} e_{2,j\\alpha}\nf_2^{(I_1+2I_n)}\\bigg)\n = - 2i \\bigg(\\frac{e_{1,j\\alpha}}{\\mu_1} C_1 + \\frac{e_{2,j\\alpha}}{\\mu_2} C_2 \\bigg).\\\\\n\\end{split}\n\\end{equation*}\n\nWe also have\n\\begin{equation*}\n\\begin{split}\n&{\\cal L}_1^3 \\phi_{33}|_{(0,0)} = 6\\cdot\n\\frac{2\\mu_1}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}} \\ov{e_{2,11}} + 6i \\chi_1\n\\cdot 2\\cdot 2i\\bigg(\\frac{\\sqrt{\\mu_1}}{\\mu_1} \\ov{e_{1,11}}\\cdot\ne_{1,33}\n+ \\frac{\\sqrt{\\mu_1}}{\\mu_2} \\ov{e_{2,11}} \\cdot e_{2,33}\\\\\n&\\ \\ + \\frac{\\sqrt{\\mu_1} i}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}}\nf_2^{(I_1+2I_n)}\\bigg) - 12 \\chi_1^2 \\cdot 2\\cdot\n2i\\bigg(-\\frac{1}{\\mu_1}e_{1,33}f_1^{(I_1+2I_n)} - \\frac{1}{\\mu_2}\ne_{2,33}f_2^{(I_1+2I_n)}\\bigg)\\\\& = \\frac{\\sqrt{\\mu_1}\ni}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)} \\chi_1} C_2\n - 2i\n\\bigg(\\frac{e_{1,33}}{\\mu_1} C_1+\\frac{e_{2,33}}{\\mu_2} C_2\\bigg).\n\\end{split}\n\\end{equation*}\n\n\\medspace\n\nWe write $B_2=(Id-G_1)B_1$ and $A_2=(Id-G_1)A_1$ so that\n\\begin{equation}\n\\label{new form b} B_2 \\ov{\\phi}(\\chi, 0)^t=A_2\n\\end{equation}\nwhere\n\\[\nG_1:=\n\\begin{pmatrix}\n0 & 0 & 0 & \\frac{2\\chi_1}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_2}{\\chi_3} & 0 & \\frac{\\chi_1}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{2\\chi_2}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{-C_1}{\\mu_1 \\chi_3} & 0 & \\frac{-C_2}{\\mu_2 \\chi_3} & 0 & 0 \\\\\n\\end{pmatrix}.\n\\]\nThen\n\\begin{equation*}\n\\begin{split}\n& B_{2,11}=B_{1,11} - \\frac{2\\chi_1}{\\chi_3} B_{1,13}=(2\\mu_{11}, 0,\n0, -\\frac{2\\chi_1}{\\chi_3}\n\\mu_{13}, 0, ..., 0),\\\\\n& B_{2,12}=B_{1,12} -\\bigg(\\frac{\\chi_2}{\\chi_3} B_{1,13}\n+ \\frac{\\chi_1}{\\chi_3} B_{1,23} \\bigg)=(0, \\mu_{12}, 0, -\\frac{\\chi_2}{\\chi_3} \\mu_{13}, 0, ..., 0, -\\frac{\\chi_1}{\\chi_3} \\mu_{23}, 0, ..., 0), \\\\\n& B_{2,22} = B_{1,22} - \\frac{2\\chi_2}{\\chi_3} B_{1,23}=(0,0,\n2\\mu_{22}, 0, ..., 0,\n-\\frac{2\\chi_2}{\\chi_3} \\mu_{23}, 0, ..., 0),\\\\\n& B_{2,13}=B_{1,13},\\\\\n& B_{2,23}=B_{1,23}, \\\\\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n& B_{2,1\\alpha}=B_{1,1\\alpha} - \\frac{\\chi_\\alpha}{\\chi_3} B_{1,13}\n=(0,0,0, -\\frac{\\chi_\\alpha}{\\chi_3} \\mu_{13}, 0, .., 0, \\mu_{1\\alpha}, 0, ..., 0),\\\\\n& B_{2,2\\alpha}=B_{1,2\\alpha} - \\frac{\\chi_\\alpha}{\\chi_3} B_{1,23}\n=(0,0,0, 0, ..., 0, -\\frac{\\chi_\\alpha}{\\chi_3} \\mu_{23}, 0, .., 0, \\mu_{2\\alpha}, 0, ..., 0),\\\\\n& B_{2,33} = B_{1,33} + \\frac{B_{1,13}}{\\mu_1 \\chi_3} C_1 + \\frac{B_{1,23}}{\\mu_2 \\chi_3} C_2\\\\\n& =\\bigg(\\frac{-C_1}{\\sqrt{\\mu_1}\\chi_1} ,\n\\frac{-C_2}{\\sqrt{\\mu_1+\\mu_2}\\chi_1}, 0,\n\\frac{C_1}{\\sqrt{\\mu_1}\\chi_3}, 0, ..., 0, \\frac{C_2}{\\sqrt{\\mu_2}\n\\chi_3}, 0, ..., 0, \\frac{\\sqrt{\\mu_1} i\nC_2}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}\\chi_1}\\bigg),\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation}\n\\begin{split}\n& A_{2,11} = 2\\mu_1 \\chi_1^2 - \\frac{2 \\chi_1}{\\chi_3} \\mu_1 \\chi_1 \\chi_3 = 0,\\\\\n& A_{2,12} = (\\mu_1+\\mu_2) \\chi_1 \\chi_2 -\n\\bigg(\\frac{\\chi_2}{\\chi_3} \\mu_1 \\chi_1\\chi_3\n+ \\frac{\\chi_1}{\\chi_3} \\mu_2 \\chi_2 \\chi_3 \\bigg) = 0,\\\\\n& A_{2,22} = 2 \\mu_2 \\chi_2^2 - \\frac{2 \\chi_2}{\\chi_3} \\mu_2 \\chi_2 \\chi_3 = 0,\\\\\n& A_{2,13} = A_{1,13},\\ \\ A_{2,23} = A_{1, 23}, \\\\\n& A_{2,1\\alpha} = \\mu_1 \\chi_1 \\chi_\\alpha - \\frac{\\chi_\\alpha}{\\chi_3} \\mu_1 \\chi_1 \\chi_3 = 0,\\\\\n& A_{2,2\\alpha} = \\mu_2 \\chi_2 \\chi_\\alpha - \\frac{\\chi_\\alpha}{\\chi_3} \\mu_2 \\chi_2 \\chi_3 = 0,\\\\\n& A_{2, 33} = - \\chi_1 C_1 - \\chi_2 C_2 + \\frac{\\mu_1 \\chi_1\n\\chi_3}{\\mu_1 \\chi_3} C_1 + \\frac{\\mu_2 \\chi_2\\chi_3}{\\mu_2 \\chi_3}\nC_2 = 0,\n\\end{split}\n\\end{equation}\nwhere $4\\le \\alpha\\le n-1$. From (\\ref{new form b}) we have\n\\begin{equation}\n\\begin{split}\\label{rel}\n&\\ov{\\phi_{11}}(\\chi, 0)=\\frac{\\chi_1}{\\chi_3} \\ov{\\phi_{13}}(\\chi,\n0),\\ \\ \\ov{\\phi_{12}}(\\chi, 0)=\\frac{\\mu_{13}\\chi_2}{\\mu_{12}\\chi_3}\n\\ov{\\phi_{13}}(\\chi, 0)\n+ \\frac{\\mu_{23}\\chi_1}{\\mu_{12}\\chi_3} \\ov{\\phi_{23}}(\\chi, 0), \\\\\n& \\ov{\\phi_{22}}(\\chi, 0)=\\frac{\\chi_2}{\\chi_3} \\ov{\\phi_{23}}(\\chi,\n0),\\ \\ \\ov{\\phi_{1\\alpha}}(\\chi, 0)=\\frac{\\chi_\\alpha}{\\chi_3}\n\\ov{\\phi_{13}}(\\chi, 0),\\ \\ \\ov{\\phi_{2\\alpha}}(\\chi,\n0)=\\frac{\\chi_\\alpha}{\\chi_3} \\ov{\\phi_{23}}(\\chi, 0),\n\\end{split}\n\\end{equation}\nwhere $4\\le \\alpha\\le n-1$. By considering $B_{2,13} \\ov{\\phi}(\\chi,\n0)^t=A_{2,13}$, we obtain\n\\begin{equation}\n\\begin{split}\n& 2i \\chi_3 e_{1,11} \\ov{\\phi_{11}} + 2i \\chi_3 e_{1,12}\n\\ov{\\phi_{12}}\n+ 2i \\chi_3 e_{1,22} \\ov{\\phi_{22}}+\\mu_{13}\\ov{\\phi_{13}}\\\\\n&+ \\sum^{n-1}_{\\alpha=3} 2i \\chi_3 e_{1,1\\alpha} \\ov{\\phi_{1\\alpha}}\n+ \\sum^{n-1}_{\\alpha=3} 2i \\chi_3 e_{1,2\\alpha} \\ov{\\phi_{2\\alpha}}\n+ 2i e_{1,33} \\chi_3 \\ov{\\phi_{33}} = \\mu_1 \\chi_1 \\chi_3.\n\\end{split}\n\\end{equation}\n\nSubstituting (\\ref{rel}) into this equation, we get\n\n\\begin{equation} \\label{2 13 eq}\n\\begin{split}\n& \\bigg[\\mu_{13} + 2i\\bigg(e_{1,11}\\chi_1 +\n\\frac{\\sqrt{\\mu_1}}{\\mu_{12}} e_{1,12} \\chi_2\n+ \\sum^{n-1}_{\\alpha=3} e_{1,1\\alpha} \\chi_\\alpha \\bigg) \\bigg] \\ov{\\phi_{13}}\\\\\n& + \\bigg(e_{1,12} \\chi_2 + \\frac{\\sqrt{\\mu_2}}{\\mu_{12}} e_{1,12}\n\\chi_1 + \\sum^{n-1}_{\\alpha=3} e_{1,2\\alpha} \\chi_\\alpha\\bigg)\n\\ov{\\phi_{23}} + 2i e_{1,33} \\chi_3 \\ov{\\phi_{33}} = \\mu_1 \\chi_1\n\\chi_3.\n\\end{split}\n\\end{equation}\n\n\\medspace\n\nWe can rewrite (\\ref{2 13 eq}) as\n\\begin{equation} \\label{e1}\n(\\mu_{13} + 2i \\sqrt{\\mu_1} E_1) \\ov{\\phi_{13}} + 2i \\sqrt{\\mu_2}\nE_2 \\ov{\\phi_{23}} + 2i e_{1,33} \\chi_3 \\ov{\\phi_{33}} = \\mu_1\n\\chi_1 \\chi_3\n\\end{equation}\nwhere\n\\begin{equation*}\nE_1= \\frac{1}{\\mu_{11}} e_{1,11}\\chi_1 + \\frac{1}{\\mu_{12}} e_{1,12}\n\\chi_2 + \\sum^{n-1}_{\\alpha=3} \\frac{1}{\\mu_{1\\alpha}} e_{1,1\\alpha}\n\\chi_\\alpha,\\ \\ E_2= \\frac{1}{\\mu_{22}} e_{1,22}\\chi_2 +\n\\frac{1}{\\mu_{12}} e_{1,12} \\chi_1 + \\sum^{n-1}_{\\alpha=3}\n\\frac{1}{\\mu_{2\\alpha}} e_{1,2\\alpha} \\chi_\\alpha.\n\\end{equation*}\n\n\\medspace\n\nSimilarly by considering $B_{2,23} \\ov{\\phi}(\\chi, 0)^t = A_{2,23}$,\nwe obtain\n\\begin{equation} \\label{e2}\n\\begin{split}\n2i \\sqrt{\\mu_1} E_3 \\ov{\\phi_{13}} + (\\mu_{23} + 2i \\sqrt{\\mu_2}\nE_4) \\ov{\\phi_{23}} + 2i e_{2,33} \\chi_3 \\ov{\\phi_{33}} = \\mu_2\n\\chi_2 \\chi_3\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation*}\nE_3= \\frac{1}{\\mu_{11}} e_{2,11}\\chi_1 + \\frac{1}{\\mu_{12}} e_{2,12}\n\\chi_2 + \\sum^{n-1}_{\\alpha=3} \\frac{1}{\\mu_{1\\alpha}} e_{2,1\\alpha}\n\\chi_\\alpha,\\ \\ E_4= \\frac{1}{\\mu_{22}} e_{2,22}\\chi_2 +\n\\frac{1}{\\mu_{12}} e_{2,12} \\chi_1 + \\sum^{n-1}_{\\alpha=3}\n\\frac{1}{\\mu_{2\\alpha}} e_{2,2\\alpha} \\chi_\\alpha.\n\\end{equation*}\n\n\\medspace\n\nSubstituting (\\ref{rel}) into $B_{2,33} \\ov{\\phi}(\\chi, 0)^t =\nA_{2,33}$, we yield\n\\begin{equation}\n\\begin{split}\n& -\\frac{C_1}{\\sqrt{\\mu_1}\\chi_1} \\frac{\\chi_1}{\\chi_3}\n\\ov{\\phi_{13}}(\\chi, 0) -\n\\frac{C_2}{\\sqrt{\\mu_1+\\mu_2}\\chi_1}\\bigg(\\frac{\\sqrt{\\mu_1}\n\\chi_2}{\\mu_{12} \\chi_3} \\ov{\\phi_{13}}(\\chi, 0)\n+ \\frac{\\sqrt{\\mu_2}\\chi_1}{\\mu_{12} \\chi_3} \\ov{\\phi_{23}}(\\chi, 0) \\bigg)\\\\\n& + \\frac{C_1}{\\sqrt{\\mu_1}\\chi_3} \\ov{\\phi_{13}}(\\chi, 0) +\n\\frac{C_2}{\\sqrt{\\mu_2} \\chi_3} \\ov{\\phi_{23}}(\\chi, 0) +\n\\frac{\\sqrt{\\mu_1} i C_2}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}\\chi_1}\n\\ov{\\phi_{33}}(\\chi, 0)=0.\n\\end{split}\n\\end{equation}\nWe notice that the polynomial $C_2\\not=0$ because $e_{2,11}\\not=0$.\nHere we used the fact that $\\phi_{33}^{(3I_1)}=\\frac{2\n\\mu_1}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}} \\ov{e_{2,11}}$ $\\not=0$ in Case A.\nThen divided by $C_2$, we obtain from above\n\\[\n\\frac{\\sqrt{\\mu_1}\\chi_2}{\\mu_1+\\mu_2} \\ov{\\phi_{13}}(\\chi, 0) -\n\\frac{\\mu_1 \\chi_1}{\\sqrt{\\mu_2} (\\mu_1+\\mu_2)} \\ov{\\phi_{23}}(\\chi,\n0) - \\frac{\\sqrt{\\mu_1} i \\chi_3}{\\sqrt{(\\mu_1+\\mu_2)\\mu_2}}\n\\ov{\\phi_{33}}(\\chi, 0) = 0,\n\\]\nand hence\n\\begin{equation} \\label{33 eq}\n\\ov{\\phi_{33}}(\\chi, 0) =\n\\frac{i}{\\sqrt{\\mu_1+\\mu_2}\\chi_3}(-\\sqrt{\\mu_2} \\chi_2\n\\ov{\\phi_{13}} + \\sqrt{\\mu_1} \\chi_1 \\ov{\\phi_{23}})(\\chi, 0).\n\\end{equation}\nThen (\\ref{e1}) and (\\ref{e2}) further take the form at $(\\chi, 0)$:\n\\begin{equation}\n\\begin{split}\n&\\bigg[\\mu_{13}+2i\\sqrt{\\mu_1} E_1 +\n2\\sqrt{\\frac{\\mu_2}{\\mu_1+\\mu_2}} e_{1,33}\n\\chi_2\\bigg]\\ov{\\phi_{13}} +\\bigg[2i \\sqrt{\\mu_2} E_2 -\n2\\sqrt{\\frac{\\mu_1}{\\mu_1+\\mu_2}} e_{1,33}\n\\chi_1\\bigg]\\ov{\\phi_{23}} =\n\\mu_1 \\chi_1 \\chi_3,\\\\\n&\\bigg[2i \\sqrt{\\mu_1} E_3 + 2\\sqrt{\\frac{\\mu_2}{\\mu_1+\\mu_2}}\ne_{2,33} \\chi_2\\bigg]\\ov{\\phi_{13}} + \\bigg[\\mu_{23}+2i\\sqrt{\\mu_2}\nE_4 - 2\\sqrt{\\frac{\\mu_1}{\\mu_1+\\mu_2}} e_{2,33}\n\\chi_1\\bigg]\\ov{\\phi_{23}} = \\mu_2 \\chi_2 \\chi_3.\n\\end{split}\n\\end{equation}\nIt implies that $\\ov{\\phi_{13}}(\\chi, 0)$ and $\\ov{\\phi_{23}}(\\chi,\n0)$ take the form $\\chi_3 \\frac{P_1^{(2)}(\\chi)}{P_2^{(2)}(\\chi)}$\nwhere $P_1^{(2)}(\\chi)$ and $P_2^{(2)}(\\chi)$ are polynomials of\ndegree 2. Substituting these forms back to (\\ref{rel}) and (\\ref{33\neq}),  we conclude that (\\ref{reduction}) is proved and hence\n$deg(F(z, 0))\\le 3$.\n This completes the proof of Proposition \\ref{propdeg} for Case A.\n Case A$'$ can be similarly proved.\n\\end{proof}\n\n\n\n\n\n\\section{Proof of Proposition \\ref{propdeg} for Case B}\n\nThis section is devoted to  the proof of Proposition \\ref{propdeg}\nfor Case B.\n\n\\begin{proof}[Proof of Proposition \\ref{propdeg} for Case B]\nIn this case, we suppose  $\\phi^{(2I_1+I_2)}_{33}\\neq 0$. Thus\n\\[\n\\frac{\\mathcal{L}_1^2\\mathcal{L}_2  g}{2i}|_{(0, 0)} =\n\\sum^{n-1}_{l=1} \\mathcal{L}_1^2\\mathcal{L}_2 f_l|_{(0, 0)}\n\\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal S}_0}\n\\mathcal{L}_1^2\\mathcal{L}_2 \\phi_{st}|_{(0, 0)}\n\\ov{\\phi_{st}(\\ov\\chi, 0)} + \\mathcal{L}_1^2\\mathcal{L}_2\n\\phi_{33}|_{(0, 0)} \\ov{\\phi_{33}(\\ov\\chi,\n    0)}.\n\\]\nWrite\n\\begin{equation}\\begin{split}\nD_1=&-4i\\chi_1\\cdot \\mu_{12}\\ov{e_{1,12}} +4i\\chi_1\\cdot\n2i\\chi_2\\cdot f_1^{(I_1+2I_n)}+(2i\\chi_1)^2\\cdot\n2f_1^{(I_2+2I_n)},\\\\\nD_2=&-4i\\chi_2\\cdot \\mu_{11}\\ov{e_{1,11}} +4i\\chi_1\\cdot\n2i\\chi_2\\cdot f_1^{(I_1+2I_n)},\\\\\nD_3=&-4i\\chi_1\\cdot \\mu_{12}\\ov{e_{2,12}} +4i\\chi_1\\cdot\n2i\\chi_2\\cdot f_2^{(I_1+2I_n)}+(2i\\chi_1)^2\\cdot\n2f_2^{(I_2+2I_n)},\\\\\nD_4=&-4i\\chi_2\\cdot \\mu_{11}\\ov{e_{2,11}} +4i\\chi_1\\cdot\n2i\\chi_2\\cdot f_2^{(I_1+2I_n)}.\n\\end{split}\\end{equation}\nA direct computation shows that $\\mathcal{L}_1^2\\mathcal{L}_2 =\n\\frac{\\p ^3}{\\p z_1^2 \\p\n    z_2}+2i\\chi_2\\frac{\\p ^3}{\\p z_1^2 \\p w}+4i\\chi_1\\frac{\\p ^3}{\\p z_1\n    \\p z_2 \\p w}+4i\\chi_1\\cdot 2i\\chi_2\\frac{\\p ^3}{\\p z_1 \\p\n    w^2}\n+(2i\\chi_1)^2\\frac{\\p ^3}{\\p z_2 \\p w^2}+(2i\\chi_1)^2\\cdot\n2i\\chi_2\\frac{\\p ^3}{ \\p w^3}$. Thus\n\\begin{equation}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_2 f_1|_{(0,0)} =\n2i\\chi_2\\cdot 2f_1^{(2I_1+I_n)}+4i\\chi_1f_1^{(I_1+I_2+I_n)}\\\\\n&\\ +4i\\chi_1\\cdot 2i\\chi_2\\cdot 2f_1^{(I_1+2I_n)}\n + (2i\\chi_1)^2\\cdot 2f_1^{(I_2+2I_n)} = D_1+D_2.\n\\end{split}\n\\end{equation}\n\\begin{equation}\\begin{split}\n&\\mathcal{L}_1^2\\mathcal{L}_2 f_2|_{(0,0)} = 2i\\chi_2\\cdot\n2f_2^{(2I_1+I_n)}+4i\\chi_1f_2^{(I_1+I_2+I_n)}\\\\\n&+4i\\chi_1\\cdot 2i\\chi_2\\cdot 2f_2^{(I_1+2I_n)}+(2i\\chi_1)^2\\cdot\n2f_2^{(I_2+2I_n)} = D_3+D_4.\n\\end{split}\\end{equation}\nAs in (\\ref{new form}), we have\n\\begin{equation}\n\\label{new formb app} {\\cal B}_1 \\ov{\\phi}(\\chi, 0)^t={\\cal A}_1.\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal B}_1:= \\begin{pmatrix}\n{\\cal L}_j{\\cal L}_k \\phi_{hl} & {\\cal L}_j{\\cal L}_k \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_k \\phi_{33} \\\\\n{\\cal L}_j{\\cal L}_\\beta \\phi_{hl} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{33}\\\\\n{\\cal L}^2_1 {\\cal L}_2\\phi_{hl} & {\\cal L}^2_1 {\\cal L}_2\n\\phi_{h\\alpha} &\n{\\cal L}^2_1 {\\cal L}_2 \\phi_{33} \\\\\n\\end{pmatrix}_{1\\leq j,k,h,l\\leq 2, {3\\leq \\alpha,\\beta\\leq n-1}} \\bigg|_{(0,0)}\n=\\begin{pmatrix}\nB_{1, jk}\\\\\nB_{1, j \\beta}\\\\\n{\\cal B}_{1, 33}\n\\end{pmatrix}.\n\\end{equation}\nand ${\\cal A}_1=(A_{1,11}, A_{1,12}, A_{1,22}, A_{1,1\\alpha},\nA_{1,2\\alpha}, {\\cal A}_{1,33})^t$ where ${\\cal A}_{1,33}=-{\\cal\nL}_1^2 {\\cal L}_2 f_1|_{(0,0)} \\chi_1 - {\\cal L}_1^2 {\\cal L}_2\nf_2|_{(0,0)}$ $\\chi_2 = -(D_1+D_2) \\chi_1  - (D_3+D_4) \\chi_2$, and\n${\\cal B}_{1,33}=({\\cal L}_1^2{\\cal L}_2  \\phi_{hl}|_{(0,0)},\\ {\\cal\nL}_1^2{\\cal L}_2 \\phi_{h\\alpha}|_{(0,0)},\\ {\\cal L}_1^2{\\cal L}_2\n\\phi_{33}|_{(0,0)})$. Recall\n$\\mathcal{L}_1^2\\mathcal{L}_2\\phi|_{(0,0)} =\n2\\phi^{(2I_1+I_2)}+2i\\chi_2\\cdot\n2\\phi^{(2I_1+I_n)}+4i\\chi_1\\phi^{(I_1+I_2+I_n)} +4i\\chi_1\\cdot\n2i\\chi_2\\cdot 2\\phi^{(I_1+2I_n)}+(2i\\chi_1)^2\\cdot\n2\\phi^{(I_2+2I_n)}$. We calculate ${\\cal B}_{1,33}$ in details as\nfollows.\n\n\\begin{equation}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_2\\phi_{11}|_{(0,0)} =  2\\cdot\n\\frac{2i}{\\sqrt{\\mu_1}}\\mu_{12}\\ov{e_{1,12}}+2i\\chi_2\\cdot\n2\\Big(2i(\\frac{\\mu_{11}}{\\mu_1}\\ov{e_{1,11}}e_{1,11}\n+\\frac{\\mu_{11}}{\\mu_2}\\ov{e_{2,11}}e_{2,11}) \\\\\n&\\ \\ \\ -\\frac{2i}{\\sqrt{\\mu_1}}f_1^{(I_1+2I_n)}\\Big)\n + 4i\\chi_1\\cdot \\Big(2i(\\frac{\\mu_{12}}{\\mu_1}\\ov{e_{1,12}}e_{1,11}\n+\\frac{\\mu_{12}}{\\mu_2}\\ov{e_{2,12}}e_{2,11})-\\frac{2i}{\\sqrt{\\mu_1}}f_1^{(I_2+2I_n)}\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot 2i\\chi_2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,11}f_1^{(I_1+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,11}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + (2i\\chi_1)^2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,11}f_1^{(I_2+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,11}f_2^{(I_2+2I_n)}\\Big)\\\\\n& =\n-\\frac{2i}{\\mu_1}e_{1,11}(D_1+D_2)-\\frac{2i}{\\mu_2}e_{2,11}(D_3+D_4)-\\frac{D_1}{\\sqrt{\\mu_1}\\chi_1},\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_2\\phi_{12}|_{(0,0)} =  2\\cdot\n\\frac{2i}{{\\mu_{12}}}(\\mu_{12}\\ov{e_{2,12}}+\\mu_{11}\\ov{e_{1,11}}\\big)\\\\\n&\\ \\ \\ + 2i\\chi_2\\cdot\n2\\Big(2i(\\frac{\\mu_{11}}{\\mu_1}\\ov{e_{1,11}}e_{1,12}\n+\\frac{\\mu_{11}}{\\mu_2}\\ov{e_{2,11}}e_{2,12})-\\frac{2i}{{\\mu_{12}}}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot\n\\Big(2i(\\frac{\\mu_{12}}{\\mu_1}\\ov{e_{1,12}}e_{1,12}\n+\\frac{\\mu_{12}}{\\mu_2}\\ov{e_{2,12}}e_{2,12})-\\frac{2i}{\\mu_{12}}(f_1^{(I_1+2I_n)}+f_2^{(I_2+2I_n)})\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot 2i\\chi_2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,12}f_1^{(I_1+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,12}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + (2i\\chi_1)^2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,12}f_1^{(I_2+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,12}f_2^{(I_2+2I_n)}\\Big)\\\\\n\\end{split}\\end{equation}\n\n\\begin{equation*}\\begin{split}\n& =\n-\\frac{2i}{{\\mu_1}}e_{1,12}(D_1+D_2)-\\frac{2i}{\\mu_2}e_{2,12}(D_3+D_4)\n-\\frac{1}{\\mu_{12}\\chi_1}D_3-\\frac{1}{\\mu_{12}\\chi_2}D_2,\n\\end{split}\\end{equation*}\n\n\\begin{equation}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_2\\phi_{22}|_{(0,0) } = 2\\cdot\n\\frac{2i}{{\\mu_{22}}}\\mu_{11}\\ov{e_{2,11}}+2i\\chi_2\\cdot 2\\cdot\n2i\\big(\\frac{\\mu_{11}}{\\mu_1}\\ov{e_{1,11}}e_{1,22}\n+\\frac{\\mu_{11}}{\\mu_2}\\ov{e_{2,11}}e_{2,22}\\big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot\n\\Big(2i(\\frac{\\mu_{12}}{\\mu_1}\\ov{e_{1,12}}e_{1,22}\n+\\frac{\\mu_{12}}{\\mu_2}\\ov{e_{2,12}}e_{2,22})-\\frac{2i}{\\sqrt{\\mu_{2}}}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot 2i\\chi_2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,22}f_1^{(I_1+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,22}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + (2i\\chi_1)^2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,22}f_1^{(I_2+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,22}f_2^{(I_2+2I_n)}\\Big)\\\\\n& =\n-\\frac{2i}{{\\mu_1}}e_{1,22}(D_1+D_2)-\\frac{2i}{{\\mu_2}}e_{2,22}(D_3+D_4)\n-\\frac{1}{\\sqrt{\\mu_{2}}\\chi_2}D_4,\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_2\\phi_{j\\alpha}|_{(0,0)} =\n2i\\chi_2\\cdot 2\\cdot\n2i(\\frac{\\mu_{11}}{\\mu_1}\\ov{e_{1,11}}e_{1,j\\alpha}\n+\\frac{\\mu_{11}}{\\mu_2}\\ov{e_{2,11}}e_{2,j\\alpha})\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot\n2i(\\frac{\\mu_{12}}{\\mu_1}\\ov{e_{1,12}}e_{1,j\\alpha}\n+\\frac{\\mu_{12}}{\\mu_2}\\ov{e_{2,12}}e_{2,j\\alpha})\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot 2i\\chi_2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,j\\alpha}f_1^{(I_1+2I_n)}\n-\\frac{2i}{{\\mu_2}}e_{2,j\\alpha}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + (2i\\chi_1)^2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,j\\alpha}f_1^{(I_2+2I_n)}\n-\\frac{2i}{{\\mu_2}}e_{2,j\\alpha}f_2^{(I_2+2I_n)}\\Big)\\\\\n& =\n-\\frac{2i}{{\\mu_1}}e_{1,j\\alpha}(D_1+D_2)-\\frac{2i}{{\\mu_2}}e_{2,j\\alpha}(D_3+D_4),\n\\end{split}\\end{equation}\nfor $j=1, 2$, and\n\\begin{equation}\\begin{split}\n&\\mathcal{L}_1^2\\mathcal{L}_2\\phi_{33}|_{(0,0)} = 2\\cdot\n\\frac{2}{\\mu_{12}}\\big(\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\mu_{12}\\ov{e_{2,12}}\n-\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\mu_{11}\\ov{e_{1,11}}\\big)\\\\\n&\\ \\ \\ +2i\\chi_2\\cdot\n2\\Big(2i(\\frac{\\mu_{11}}{\\mu_1}\\ov{e_{1,11}}e_{1,33}\n+\\frac{\\mu_{11}}{\\mu_2}\\ov{e_{2,11}}e_{2,33})-\\frac{2}{{\\mu_{12}}}\n\\sqrt{\\frac{\\mu_1}{\\mu_2}}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot\n\\Big(2i(\\frac{\\mu_{12}}{\\mu_1}\\ov{e_{1,12}}e_{1,33}\n+\\frac{\\mu_{12}}{\\mu_2}\\ov{e_{2,12}}e_{2,33})-\\frac{2}{\\mu_{12}}(\\sqrt{\\frac{\\mu_1}{\\mu_2}}f_2^{(I_2+2I_n)}\n-\\sqrt{\\frac{\\mu_2}{\\mu_1}}f_1^{(I_1+2I_n)})\\Big)\\\\\n&\\ \\ \\ + 4i\\chi_1\\cdot 2i\\chi_2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,33}f_1^{(I_1+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,33}f_2^{(I_1+2I_n)}\\Big)\\\\\n&\\ \\ \\ + (2i\\chi_1)^2\\cdot\n2\\Big(-\\frac{2i}{{\\mu_1}}e_{1,33}f_1^{(I_2+2I_n)}-\\frac{2i}{{\\mu_2}}e_{2,33}f_2^{(I_2+2I_n)}\\Big)\\\\\n\\end{split}\\end{equation}\n\\begin{equation*}\\begin{split}\n=&\\ \\ \\\n-\\frac{2i}{{\\mu_1}}e_{1,33}(D_1+D_2)-\\frac{2i}{{\\mu_2}}e_{2,33}(D_3+D_4)\n+\\frac{i}{\\mu_{12}\\chi_1} \\sqrt{\\frac{\\mu_1}{\\mu_2}}\nD_3-\\frac{i}{\\mu_{12}\\chi_2} \\sqrt{\\frac{\\mu_2}{\\mu_1}}  D_2.\n\\end{split}\\end{equation*}\n\n\\medspace\n\nWe write ${\\cal B}_2=(Id-{\\cal G}_1){\\cal B}_1$ and ${\\cal\nA}_2=(Id-{\\cal G}_1){\\cal A}_1$ so that\n\\[\n{\\cal B}_2 \\ov{\\phi}(\\chi, 0)^t = {\\cal A}_2\n\\]\nwhere\n\\[\n{\\cal G}_1:=\n\\begin{pmatrix}\n0 & 0 & 0 & \\frac{2\\chi_1}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_2}{\\chi_3} & 0 & \\frac{\\chi_1}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{2\\chi_2}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{D_1+D_2}{\\mu_1 \\chi_3} & 0 & -\\frac{D_3+D_4}{\\mu_2 \\chi_3} & 0 & 0 \\\\\n\\end{pmatrix}.\n\\]\nBy the construction of ${\\cal B}_2$ and ${\\cal A}_2$, we see that\n(\\ref{rel})(\\ref{e1}) and (\\ref{e2}) still hold. We further\ncalculate:\n\\begin{equation}\\begin{split}\n{\\cal B}_{2,33} & ={\\cal B}_{1,33}+\\frac{B_{1,13}}{\\mu_1\\chi_3}(D_1+D_2)+\\frac{B_{1,23}}{\\mu_2\\chi_3}(D_3+D_4)\\\\\n& =\n\\Big(-\\frac{D_1}{\\sqrt{\\mu_1}\\chi_1},-\\frac{D_3}{\\mu_{12}\\chi_1}-\\frac{D_2}{\\mu_{12}\\chi_2},\n-\\frac{D_4}{\\sqrt{\\mu_2}\\chi_2},\\frac{D_1+D_2}{\\sqrt{\\mu_1}\\chi_3},0,\\cdots,0,\\\\\n&\\frac{D_3+D_4}{\\sqrt{\\mu_2}\\chi_3},\n0,\\cdots,0,\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{D_3}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{D_2}{\\chi_2}\\Big).\n\\end{split}\\end{equation}\nand\n\\begin{equation*}\\begin{split}\n& {\\cal A}_{2,33} =\n-\\chi_1(D_1+D_2)-\\chi_2(D_3+D_4)+\\frac{\\mu_1\\chi_1\\chi_3}{\\mu_1\\chi_3}(D_1+D_2)\n+\\frac{\\mu_2\\chi_2\\chi_3}{\\mu_2\\chi_3}(D_3+D_4) = 0.\n\\end{split}\\end{equation*}\n\n\\medspace\n\nWe turn to ${\\cal B}_{2,33} \\ov\\phi(\\chi, 0)^t = {\\cal A}_{2,33}$ to\nhave\n\\begin{equation}\\begin{split}\n&-\\frac{D_1}{\\sqrt{\\mu_1}\\chi_1}\\ov{\\phi}_{11}-\\Big(\\frac{D_3}{\\mu_{12}\\chi_1}+\\frac{D_2}{\\mu_{12}\\chi_2}\\Big)\\ov{\\phi}_{12}\n-\\frac{D_4}{\\sqrt{\\mu_2}\\chi_2}\\ov{\\phi}_{22}\\\\\n&+\\frac{D_1+D_2}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}+\\frac{D_3+D_4}{\\sqrt{\\mu_2}\\chi_3}\n\\ov{\\phi}_{23}+\\Big(\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{D_3}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{D_2}{\\chi_2}\\Big)\\ov{\\phi}_{33}=0.\n\\end{split}\\end{equation}\nSubstituting (\\ref{rel}) to this equation, we obtain\n\n\\begin{equation}\\begin{split}\n&-\\frac{D_1}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}-\\Big(\\frac{D_3}{\\mu_{12}\\chi_1}+\\frac{D_2}{\\mu_{12}\\chi_2}\\Big)\\cdot\n\\Big(\\frac{\\mu_{13}\\chi_2}{\\mu_{12}\\chi_3}\\ov{\\phi_{13}}+\\frac{\\mu_{23}\\chi_1}{\\mu_{12}\\chi_3}\\ov{\\phi_{23}}\\Big)\n-\\frac{D_4}{\\sqrt{\\mu_2}\\chi_3}\\ov{\\phi}_{23}\\\\\n&+\\frac{D_1+D_2}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}+\\frac{D_3+D_4}{\\sqrt{\\mu_2}\\chi_3}\n\\ov{\\phi}_{23}+\\Big(\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{D_3}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{D_2}{\\chi_2}\\Big)\\ov{\\phi}_{33}=0.\n\\end{split}\\end{equation}\nA quick simplification gives\n\\begin{equation}\\begin{split}\n&\n\\frac{\\mu_2\\chi_1D_2-\\mu_1\\chi_2D_3}{\\sqrt{\\mu_1}(\\mu_1+\\mu_2)\\chi_1\\chi_3}\\ov{\\phi}_{13}\n-\\frac{\\mu_2\\chi_1D_2-\\mu_1\\chi_2D_3}{\\sqrt{\\mu_2}(\\mu_1+\\mu_2)\\chi_2\\chi_3}\\ov{\\phi}_{23}\n-\\frac{i({\\mu_2\\chi_1D_2-\\mu_1\\chi_2D_3})}{\\sqrt{\\mu_1\\mu_2(\\mu_1+\\mu_2)}\\chi_1\\chi_2}\\ov{\\phi}_{33}\n=0.\n\\end{split}\\end{equation}\nSince $\\phi^{(2I_1+I_2)}_{33}\\neq 0$ in Case B, the polynomial\n$\\mu_2\\chi_1D_2-\\mu_1\\chi_2D_3\\neq 0$ because of (\\ref{phi 1}). Thus\n\\begin{equation}\\begin{split}\n& \\frac{1}{\\sqrt{\\mu_1}(\\mu_1+\\mu_2)\\chi_1\\chi_3}\\ov{\\phi}_{13}\n-\\frac{1}{\\sqrt{\\mu_2}(\\mu_1+\\mu_2)\\chi_2\\chi_3}\\ov{\\phi}_{23}\n-\\frac{i}{\\sqrt{\\mu_1\\mu_2(\\mu_1+\\mu_2)}\\chi_1\\chi_2}\\ov{\\phi}_{33}\n=0.\n\\end{split}\\end{equation}\nHence we obtain the same formula (\\ref{33 eq}). As in case A, we\nshow $deg(F)\\le 3$. The proof for the case B$'$ is similar to the\ncase B.\n\\end{proof}\n\n\n\\section{Proof of Proposition \\ref{propdeg} for Case C}\n\nIn this section, we will prove Proposition \\ref{propdeg} for Case C.\n\n\\begin{proof}[Proof of Proposition \\ref{propdeg} for Case C]\nIn Case C, we suppose  $\\phi^{(I_1+I_2+I_j)}_{33}\\neq 0$. Then\n\\[\n\\frac{{\\cal L}_1 {\\cal L}_2 {\\cal L}_j  g}{2i}|_{(0, 0)} =\n\\sum^{n-1}_{l=1} {\\cal L}_1 {\\cal L}_2 {\\cal L}_j f_l|_{(0, 0)}\n\\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal S}_0} {\\cal L}_1 {\\cal\nL}_2 {\\cal L}_j \\phi_{st}|_{(0, 0)} \\ov{\\phi_{st}(\\ov\\chi, 0)}\n\\]\n\\[\n + {\\cal L}_1 {\\cal L}_2 {\\cal L}_j \\phi_{33}|_{(0, 0)} \\ov{\\phi_{33}(\\ov\\chi, 0)}.\n\\]\n\nRecall ${\\cal L}_1 \\mathcal{L}_2\\mathcal{L}_j = \\frac{\\p^3}{\\p z_1\nz_2 \\p z_j} + 2i\\chi_1 \\frac{\\p^3}{\\p z_2 z_j \\p w} +\n2i\\chi_2\\frac{\\p^3}{\\p z_1 \\p z_j \\p w} + 2i\\chi_j\\frac{\\p^3}{\\p z_1\n\\p z_2 \\p w} - 4 \\chi_2 \\chi_j \\frac{\\p^3}{\\p z_1 \\p w^2} - 4 \\chi_1\n\\chi_j \\frac{\\p^3}{\\p z_2 \\p w^2} - 4 \\chi_1 \\chi_2 \\frac{\\p^3}{\\p\nz_j \\p w^2} - 8 i \\chi_1\\chi_2 \\chi_j\\frac{\\p^3}{ \\p w^3}$. Then\n\\begin{equation*}\\begin{split}\n& {\\cal L}_1{\\cal L}_2{\\cal L}_j f_1|_{(0,0)}= 2i\\chi_1\nf_1^{(I_2+I_j+I_n)} + 2i\\chi_2 f_1^{(I_1+I_j+I_n)}\n+ 2i\\chi_j f_1^{(I_1+I_2+I_n)} \\\\\n& \\ \\ - 8 \\chi_2 \\chi_j f_1^{(I_1+2I_n)} - 8 \\chi_1 \\chi_j\nf_1^{(I_2+2I_n)} = \\mathtt{D}_1 + \\mathtt{D}_2 + \\mathtt{D}_3,\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\\begin{split}\n& {\\cal L}_1{\\cal L}_2{\\cal L}_j f_2|_{(0,0)}= 2i\\chi_1\nf_2^{(I_2+I_j+I_n)} + 2i\\chi_2 f_2^{(I_1+I_j+I_n)}\n+ 2i\\chi_j f_2^{(I_1+I_2+I_n)} \\\\\n& \\ \\ - 8 \\chi_2 \\chi_j f_2^{(I_1+2I_n)} - 8 \\chi_1 \\chi_j\nf_2^{(I_2+2I_n)} =\\mathtt{D}_4 + \\mathtt{D}_5 + \\mathtt{D}_6\n\\end{split}\n\\end{equation*}\nwhere\n\\[\n\\begin{cases}\n\\mathtt{D}_1 = - 2i \\chi_1 \\sqrt{\\mu_2} \\ov{e_{1,2j}} - 4 \\chi_1 \\chi_j f_1^{(I_2+2I_n)},\\\\\n\\mathtt{D}_2 = -2i \\chi_2 \\sqrt{\\mu_1} \\ov{e_{1,1j}} - 4 \\chi_2 \\chi_j f_1^{(I_1+2I_n)},\\\\\n\\mathtt{D}_3 = - 2i \\chi_j \\sqrt{\\mu_1+\\mu_2} \\ov{e_{1,12}} - 4\n\\chi_1 \\chi_j f_1^{(I_2+2I_n)}\n- 4\\chi_2\\chi_j f_1^{(I_1+2I_n)}, \\\\\n\\end{cases}\n\\]\n\\[\n\\begin{cases}\n\\mathtt{D}_4 = - 2i \\chi_1 \\sqrt{\\mu_2} \\ov{e_{2,2j}} - 4 \\chi_1 \\chi_j f_2^{(I_2+2I_n)},\\\\\n\\mathtt{D}_5 = -2i \\chi_2 \\sqrt{\\mu_1} \\ov{e_{2,1j}} - 4 \\chi_2 \\chi_j f_2^{(I_1+2I_n)} ,\\\\\n\\mathtt{D}_6 = - 2i \\chi_j \\sqrt{\\mu_1+\\mu_2} \\ov{e_{2,12}} - 4\n\\chi_1 \\chi_j f_2^{(I_2+2I_n)}\n- 4\\chi_2\\chi_j f_2^{(I_1+2I_n)}.\\\\\n\\end{cases}\n\\]\n\nAs in (\\ref{new form}), we have\n\\begin{equation}\n\\label{new formbbb} \\mathtt{B}_1 \\ov{\\phi}(\\chi, 0)^t=\\mathtt{A}_1.\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathtt{B}_1:= \\begin{pmatrix}\n{\\cal L}_j{\\cal L}_k \\phi_{hl} & {\\cal L}_j{\\cal L}_k \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_k \\phi_{33} \\\\\n{\\cal L}_j{\\cal L}_\\beta \\phi_{hl} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{33}\\\\\n{\\cal L}_1 {\\cal L}_2 {\\cal L}_j\\phi_{hl} & {\\cal L}^2_1 {\\cal L}_2\n\\phi_{h\\alpha} &\n{\\cal L}_1 {\\cal L}_2 {\\cal L}_j\\phi_{33} \\\\\n\\end{pmatrix}_{1\\leq j,k,h,l\\leq 2, {3\\leq \\alpha,\\beta\\leq n-1}} \\bigg|_{(0,0)}\n=\\begin{pmatrix}\nB_{1, jk}\\\\\nB_{1, j \\beta}\\\\\n\\mathtt{B}_{1, 33}\n\\end{pmatrix}.\n\\end{equation}\nand $\\mathtt{A}_1=(A_{1,11}, A_{1,12}, A_{1,22}, A_{1,1\\alpha},\nA_{1,2\\alpha}, \\mathtt{A}_{1,33})^t$ where\n\n\\noindent $\\mathtt{A}_{1,33}=-{\\cal L}_1 {\\cal L}_2 {\\cal L}_j\nf_1|_{(0,0)} \\chi_1 - {\\cal L}_1 {\\cal L}_2 {\\cal L}_j f_2|_{(0,0)}$\n$\\chi_2 = -(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3) \\chi_1  -\n(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) \\chi_2$, and\n\n\\noindent $\\mathtt{B}_{1,33}=({\\cal L}_1 {\\cal L}_2  {\\cal L}_j\n\\phi_{hl}|_{(0,0)},\\ {\\cal L}_1 {\\cal L}_2 {\\cal L}_j\n\\phi_{h\\alpha}|_{(0,0)},\\  {\\cal L}_1 {\\cal L}_2 {\\cal L}_j\n\\phi_{33}|_{(0,0)})$. Recall ${\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j\n\\phi|_{(0,0)} = \\phi^{(I_1+I_2+I_j)}$ $+ 2i\\chi_1\n\\phi^{(I_2+I_j+I_n)} + 2i\\chi_2 \\phi^{(I_1+I_j+I_n)} + 2i\\chi_j\n\\phi^{(I_1+I_2+I_n)} - 8 \\chi_2 \\chi_j \\phi^{(I_1+2I_n)} - 8 \\chi_1\n\\chi_j \\phi^{(I_2+2I_n)}$. Then\n\\begin{equation}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{11}|_{(0,0)} =\n\\frac{2i\\sqrt{\\mu_2}}{\\sqrt{\\mu_1}} \\ov{e_{1,2j}} + 2i\\chi_1\n\\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,11}\n+ \\frac{2i}{\\mu_2} \\mu_{2j}\\ov{e_{2,2j}}e_{2,11}\\bigg) \\\\\n& + 2i\\chi_2 \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,1j}} e_{1,11}\n+ \\frac{2i}{\\mu_2} \\mu_{1j}\\ov{e_{2,1j}}e_{2,11}\\bigg) \\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,11}\n+ \\frac{2i}{\\mu_2} \\mu_{12}\\ov{e_{2,12}}e_{2,11}\n- \\frac{2i}{\\sqrt{\\mu_1}} f_1^{(I_2+2I_n)}\\bigg) \\\\\n& - 8 \\chi_2 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,11}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,11}\\bigg)\\\\\n& - 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)}\ne_{1,11}\n- \\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,11}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,11}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3) - \\frac{2i}{\\mu_2} e_{2,11}\n(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) -\n\\frac{1}{\\sqrt{\\mu_1}\\chi_1} \\mathtt{D}_1,\n\\end{split}\\end{equation}\n\n\\begin{equation*}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{12}|_{(0,0)} = \\frac{2 i\n}{\\sqrt{\\mu_1+\\mu_2}}\n\\Big(\\sqrt{\\mu_2}\\ov{e_{2,2j}}+\\sqrt{\\mu_1}\\ov{e_{1,1j}}\\Big)\n + 2i\\chi_1  \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,12}\\\\\n\\end{split}\\end{equation*}\n\\begin{equation}\\begin{split}\n&\\ \\ \\ + \\frac{2i}{\\mu_2} \\mu_{2j} \\ov{e_{2,2j}} e_{2,12}\\bigg) +\n2i\\chi_2  \\bigg(\\frac{2i}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}} e_{1,12}\n+ \\frac{2i}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}} e_{2,12}\\bigg)\\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,12}\n+ \\frac{2i}{\\mu_2} \\mu_{12} \\ov{e_{2,12}} e_{2,12}\n- \\frac{2i}{\\sqrt{\\mu_1+\\mu_2}}f_2^{(I_2+2I_n)} - \\frac{2i}{\\sqrt{\\mu_1+\\mu_2}}f_1^{(I_1+2I_n)}\\bigg)\\\\\n& - 8 \\chi_2 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,12}\n-\\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,12}\\bigg)\\\\\n& - 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)}\ne_{1,12}\n-\\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,12}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,12}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3)\n - \\frac{2i}{\\mu_2} e_{2,12} (\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6)\n - \\frac{1}{\\sqrt{\\mu_1+\\mu_2}\\chi_2} \\mathtt{D}_2\n - \\frac{1}{\\sqrt{\\mu_1+\\mu_2} \\chi_1} \\mathtt{D}_4,\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{22}|_{(0,0)} =\n\\frac{2i\\sqrt{\\mu_1}}{\\sqrt{\\mu_2}} \\ov{e_{2,1j}} + 2i\\chi_1\n\\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,22}\n+ \\frac{2i}{\\mu_2} \\mu_{2j}\\ov{e_{2,2j}}e_{2,22}\\bigg) \\\\\n& + 2i\\chi_2 \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,1j}} e_{1,22}\n+ \\frac{2i}{\\mu_2} \\mu_{1j}\\ov{e_{2,1j}}e_{2,22}\\bigg) \\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,22}\n+ \\frac{2i}{\\mu_2} \\mu_{12}\\ov{e_{2,12}}e_{2,22}\n- \\frac{2i}{\\sqrt{\\mu_2}} f_2^{(I_1+2I_n)}\\bigg) \\\\\n&\n- 8 \\chi_2 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)} e_{1,22} - \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,22}\\bigg)\\\\\n&\n- 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)} e_{1,22} - \\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,22}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,12}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3) - \\frac{2i}{\\mu_2} e_{2,12}\n(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) -\n\\frac{1}{\\sqrt{\\mu_2}\\chi_2} \\mathtt{D}_5,\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{1k}|_{(0,0)} =\n\\frac{2i\\sqrt{\\mu_1+\\mu_2}}{\\sqrt{\\mu_1}} \\delta_{jk}\\ov{e_{1,12}} \\\\\n& + 2i\\chi_1 \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,1k}\n+ \\frac{2i}{\\mu_2} \\mu_{2j} \\ov{e_{2,2j}}e_{2,1k}\n-\\frac{2i}{\\sqrt{\\mu_1}}\\delta_{jk} f_1^{(I_2+2I_n)}\\bigg)\\\\\n& + 2i\\chi_2 \\bigg(\\frac{2i}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}} e_{1,1k}\n+ \\frac{2i}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}}e_{2,1k}\n-\\frac{2i}{\\sqrt{\\mu_1}}\\delta_{jk} f_1^{(I_1+2I_n)}\\bigg)\\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,1k}\n+ \\frac{2i}{\\mu_2} \\mu_{12} \\ov{e_{2,12}}e_{2,1k}\\bigg)\\\\\n& - 8 \\chi_2 \\chi_j\\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)} e_{1,1k}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,1k}\\bigg)\\\\\n\\end{split}\\end{equation}\n\n\\begin{equation*}\\begin{split}\n&- 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)} e_{1,1k}\n- \\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,1k}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,1k}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3) - \\frac{2i}{\\mu_2} e_{2,1k}\n(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) -\n\\frac{1}{\\sqrt{\\mu_1}\\chi_j} \\delta_{jk}\\mathtt{D}_3,\n\\end{split}\\end{equation*}\n\n\n\\begin{equation}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{2k}|_{(0,0)} =\n\\frac{2i\\sqrt{\\mu_1+\\mu_2}}{\\sqrt{\\mu_2}}\\delta_{jk} \\ov{e_{2,12}} \\\\\n& + 2i\\chi_1 \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,2k}\n+ \\frac{2i}{\\mu_2} \\mu_{2j} \\ov{e_{2,2j}}e_{2,2k}\n-\\frac{2i}{\\sqrt{\\mu_2}}\\delta_{jk} f_2^{(I_2+2I_n)}\\bigg)\\\\\n& + 2i\\chi_2 \\bigg(\\frac{2i}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}} e_{1,2k}\n+ \\frac{2i}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}}e_{2,2k}\n-\\frac{2i}{\\sqrt{\\mu_2}} \\delta_{jk}f_2^{(I_1+2I_n)}\\bigg)\\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,2k}\n+ \\frac{2i}{\\mu_2} \\mu_{12} \\ov{e_{2,12}}e_{2,2k}\\bigg)\\\\\n& - 8 \\chi_2 \\chi_j\\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)} e_{1,2k}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,2k}\\bigg)\\\\\n&- 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)} e_{1,2k}\n- \\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,2k}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,2k}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3) - \\frac{2i}{\\mu_2} e_{2,2k}\n(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) -\n\\frac{1}{\\sqrt{\\mu_2}\\chi_j} \\delta_{jk}\\mathtt{D}_6,\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n& {\\cal L}_1 \\mathcal{L}_2 {\\cal L}_j \\phi_{33}|_{(0,0)}\n= \\frac{2}{\\sqrt{\\mu_1+\\mu_2}} \\Big(\\sqrt{\\mu_1}\\ov{e_{2,2j}}-\\sqrt{\\mu_2}\\ov{e_{1,1j}}\\Big) \\\\\n& + 2i\\chi_1 \\bigg(\\frac{2i}{\\mu_1} \\mu_{2j} \\ov{e_{1,2j}} e_{1,33}\n+\\frac{2i}{\\mu_2} \\mu_{2j} \\ov{e_{2,2j}} e_{2,33} \\bigg)\\\\\n& + 2i\\chi_2 \\bigg(\\frac{2i}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}} e_{1,33}\n+\\frac{2i}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}} e_{2,33} \\bigg)\\\\\n& + 2i\\chi_j \\bigg(\\frac{2i}{\\mu_1} \\mu_{12} \\ov{e_{1,12}} e_{1,33}\n+\\frac{2i}{\\mu_2} \\mu_{12} \\ov{e_{2,12}} e_{2,33} \\\\\n&\\ \\ \\ \\ \\ - \\frac{2}{\\sqrt{\\mu_1 \\mu_2(\\mu_1+\\mu_2)}}\n\\Big(\\mu_1f_2^{(I_2+2I_n)}\n-\\mu_2 f_1^{(I_1+2I_n)}\\Big)\\bigg)\\\\\n& - 8 \\chi_2 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,33}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,33}\\bigg)\\\\\n\\end{split}\\end{equation}\n\\begin{equation*}\\begin{split}\n& - 8 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_2+2I_n)}\ne_{1,33}\n- \\frac{2i}{\\mu_2} f_2^{(I_2+2I_n)} e_{2,33}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1} e_{1,33}\n(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3)\n- \\frac{2i}{\\mu_2} e_{2,33} (\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6)\\\\\n& + \\frac{i \\sqrt{\\mu_1}}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}\\chi_1}\n\\mathtt{D}_4 - \\frac{i\n\\sqrt{\\mu_2}}{\\sqrt{\\mu_1(\\mu_1+\\mu_2)}\\chi_2} \\mathtt{D}_2.\n\\end{split}\n\\end{equation*}\n\nWe write $\\mathtt{B}_2=(Id - \\mathtt{G}_1)\\mathtt{B}_1$ and\n$\\mathtt{A}_2=(Id-\\mathtt{G}_1) \\mathtt{A}_1$ so that\n\\[\n\\mathtt{B}_2 \\ov\\phi(\\chi, 0)^t = \\mathtt{A}_2\n\\]\nwhere\n\\[\n\\mathtt{G}_1:=\n\\begin{pmatrix}\n0 & 0 & 0 & \\frac{2\\chi_1}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_2}{\\chi_3} & 0 & \\frac{\\chi_1}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{2\\chi_2}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3}{\\mu_1\n\\chi_3} & 0\n& -\\frac{\\mathtt{D}_4+\\mathtt{D}_5 + \\mathtt{D}_6}{\\mu_2 \\chi_3} & 0 & 0 \\\\\n\\end{pmatrix}.\n\\]\nBy the construction of $\\mathtt{B}_2$ and $\\mathtt{A}_2$, we see\nthat (\\ref{rel})(\\ref{e1}) and (\\ref{e2}) still hold. We further\ncalculate:\n\\begin{equation}\\begin{split}\n\\mathtt{B}_{2,33} & =\n\\mathtt{B}_{1,33}+\\frac{B_{1,13}}{\\mu_1\\chi_3}(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3)\n+\\frac{B_{1,23}}{\\mu_2\\chi_3}(\\mathtt{D}_4+\\mathtt{D}_5 +\\mathtt{D}_6)\\\\\n& = \\Big(-\\frac{\\mathtt{D}_1}{\\sqrt{\\mu_1}\\chi_1},\\\n-\\frac{\\mathtt{D}_4}{\\mu_{12}\\chi_1} -\n\\frac{\\mathtt{D}_2}{\\mu_{12}\\chi_2},\\\n-\\frac{\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_2}, \\\n\\frac{\\mathtt{D}_1+\\mathtt{D}_2}{\\sqrt{\\mu_1}\\chi_3}, 0,\\cdots,0,\\\\\n&\\frac{\\mathtt{D}_4+\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_3},0, \\cdots,0,\n\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{\\mathtt{D}_4}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{\\mathtt{D}_2}{\\chi_2}\\Big).\n\\end{split}\\end{equation}\nand\n\\begin{equation}\\begin{split}\n& \\mathtt{A}_{2,33}\n= -\\chi_1(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3)-\\chi_2(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6) \\\\\n&\\ \\ +\n\\frac{\\mu_1\\chi_1\\chi_3}{\\mu_1\\chi_3}(\\mathtt{D}_1+\\mathtt{D}_2+\\mathtt{D}_3)\n+\\frac{\\mu_2\\chi_2\\chi_3}{\\mu_2\\chi_3}(\\mathtt{D}_4+\\mathtt{D}_5+\\mathtt{D}_6)\n= 0.\n\\end{split}\\end{equation}\n\n\\medspace\n\nWe turn to $\\mathtt{B}_{2,33} \\ov\\phi(\\chi, 0)^t =\n\\mathtt{A}_{2,33}$ to have\n\\begin{equation}\\begin{split}\n&-\\frac{\\mathtt{D}_1}{\\sqrt{\\mu_1}\\chi_1}\\ov{\\phi}_{11}\n-\\Big(\\frac{\\mathtt{D}_4}{\\mu_{12}\\chi_1}+\\frac{\\mathtt{D}_2}{\\mu_{12}\\chi_2}\\Big)\\ov{\\phi}_{12}\n-\\frac{\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_2}\\ov{\\phi}_{22}\\\\\n&+\\frac{\\mathtt{D}_1+\\mathtt{D}_2}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}\n+\\frac{\\mathtt{D}_4+\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_3}\\ov{\\phi}_{23}\n+\\Big(\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{\\mathtt{D}_4}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{\\mathtt{D}_2}{\\chi_2}\\Big)\\ov{\\phi}_{33}=0.\n\\end{split}\\end{equation}\nSubstituting (\\ref{rel}) to this equation, we obtain\n\\begin{equation}\\begin{split}\n&-\\frac{\\mathtt{D}_1}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}\n-\\Big(\\frac{\\mathtt{D}_4}{\\mu_{12}\\chi_1}+\\frac{\\mathtt{D}_2}{\\mu_{12}\\chi_2}\\Big)\\cdot\n\\Big(\\frac{\\mu_{13}\\chi_2}{\\mu_{12}\\chi_3}\\ov{\\phi_{13}}\n+\\frac{\\mu_{23}\\chi_1}{\\mu_{12}\\chi_3}\\ov{\\phi_{23}}\\Big)\n-\\frac{\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_3}\\ov{\\phi}_{23}\\\\\n&+\\frac{\\mathtt{D}_1+\\mathtt{D}_2}{\\sqrt{\\mu_1}\\chi_3}\\ov{\\phi}_{13}\n+\\frac{\\mathtt{D}_4+\\mathtt{D}_5}{\\sqrt{\\mu_2}\\chi_3}\\ov{\\phi}_{23}\n+\\Big(\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_1}{\\mu_2}}\\frac{\\mathtt{D}_4}{\\chi_1}\n-\\frac{i}{\\mu_{12}}\\sqrt{\\frac{\\mu_2}{\\mu_1}}\\frac{\\mathtt{D}_2}{\\chi_2}\\Big)\\ov{\\phi}_{33}=0.\n\\end{split}\\end{equation}\nA quick simplification gives\n\\begin{equation}\\begin{split}\n&-\\frac{\\mu_1\\chi_2 \\mathtt{D}_4-\\mu_2\\chi_1\n\\mathtt{D}_2}{\\sqrt{\\mu_1}(\\mu_1+\\mu_2)\\chi_1\\chi_3} \\ov{\\phi}_{13}\n+\\frac{\\mu_1\\chi_2 \\mathtt{D}_4-\\mu_2\\chi_1\n\\mathtt{D}_2}{\\sqrt{\\mu_2}(\\mu_1+\\mu_2)\\chi_2\\chi_3} \\ov{\\phi}_{23}\n+\\frac{i({\\mu_1\\chi_2 \\mathtt{D}_4-\\mu_2\\chi_1\n\\mathtt{D}_2})}{\\sqrt{\\mu_1\\mu_2(\\mu_1+\\mu_2)}\n    \\chi_1\\chi_2}\\ov{\\phi}_{33}\n=0.\n\\end{split}\\end{equation}\nSince $\\phi_{33}^{(I_1+I_2+I_j)}\\not=0$ in Case C, the polynomial\n$\\mu_2\\chi_1 \\mathtt{D}_2-\\mu_1\\chi_2 \\mathtt{D}_4\\neq 0$ because of\n(\\ref{phi 1}). Thus\n\\begin{equation}\\begin{split}\n&-\\frac{1}{\\sqrt{\\mu_1}(\\mu_1+\\mu_2)\\chi_1\\chi_3}\\ov{\\phi}_{13}\n+\\frac{1}{\\sqrt{\\mu_2}(\\mu_1+\\mu_2)\\chi_2\\chi_3}\\ov{\\phi}_{23}\n+\\frac{i}{\\sqrt{\\mu_1\\mu_2(\\mu_1+\\mu_2)}\\chi_1\\chi_2}\\ov{\\phi}_{33}\n=0.\n\\end{split}\\end{equation}\nHence we obtain the same formula (\\ref{33 eq}). As in Case A, we can\nfurther get $deg(F)\\le 3$.\n\\end{proof}\n\n\\medspace\n\n\\section{Proof of Proposition \\ref{propdeg} for Case D}\n\nIn this section, we will prove Proposition \\ref{propdeg} for Case D.\n\n\\begin{proof}[Proof of Proposition \\ref{propdeg} for Case D] In this case, we suppose  $\\phi^{(2I_1+I_j)}_{33}\\neq 0$. Then\n\\[\n\\frac{\\mathcal{L}_1^2\\mathcal{L}_j  g}{2i}|_{(0, 0)} =\n\\sum^{n-1}_{l=1} \\mathcal{L}_1^2\\mathcal{L}_j f_l|_{(0, 0)}\n\\ov{f_l(\\ov\\chi, 0)} + \\sum_{(s, t)\\in {\\cal S}_0}\n\\mathcal{L}_1^2\\mathcal{L}_j \\phi_{st}|_{(0, 0)}\n\\ov{\\phi_{st}(\\ov\\chi, 0)} + \\mathcal{L}_1^2\\mathcal{L}_j\n\\phi_{33}|_{(0, 0)} \\ov{\\phi_{33}(\\ov\\chi,\n    0)}.\n\\]\nWrite\n\\begin{equation}\\begin{split}\n\\mathfrak{D}_1 = & - 4i\\chi_1 \\mu_{1j}\\ov{e_{1,1j}} - 8 \\chi_1\n\\chi_j  f_1^{(I_1+2I_n)},\\ \\ \\mathfrak{D}_2 =  - 4i\\chi_j\n\\mu_{11}\\ov{e_{1,11}} - 8 \\chi_1 \\chi_j\nf_1^{(I_1+2I_n)},\\\\\n\\mathfrak{D}_3 = & - 4i\\chi_1  \\mu_{1j} \\ov{e_{2,1j}} - 8 \\chi_1\n\\chi_j f_2^{(I_1+2I_n)},\\ \\ \\mathfrak{D}_4 = -4i\\chi_j\n\\mu_{11}\\ov{e_{2,11}} - 8 \\chi_1 \\chi_j\n f_2^{(I_1+2I_n)}.\n\\end{split}\\end{equation}\nHere we used the fact that $f_1^{(I_j+2I_n)}=f_2^{(I_j+2I_n)}=0$ due\nto Theorem 2.1. A direct computation shows that\n$\\mathcal{L}_1^2\\mathcal{L}_j = \\frac{\\p^3}{\\p z_1^2 \\p z_j} +\n2i\\chi_j\\frac{\\p^3}{\\p z_1^2 \\p w} + 4i\\chi_1\\frac{\\p^3}{\\p z_1\\p\nz_j \\p w} - 8 \\chi_1 \\chi_j\\frac{\\p^3}{\\p z_1 \\p w^2} - 4\n\\chi_1^2\\frac{\\p^3}{\\p z_j \\p w^2} - 8 i \\chi_1^2 \\chi_j\\frac{\\p^3}{\n\\p w^3}$. Thus\n\\begin{equation}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j f_1|_{(0,0)} = 2i\\chi_j\\cdot 2\nf_1^{(2I_1+I_n)} + 4i\\chi_1 f_1^{(I_1+I_j+I_n)} - 8 \\chi_1 \\chi_j\n\\cdot 2 f_1^{(I_1+2I_n)}  = \\mathfrak{D}_1 + \\mathfrak{D}_2.\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j f_2|_{(0,0)} = 2i\\chi_j\\cdot 2\nf_2^{(2I_1+I_n)} + 4i\\chi_1 f_2^{(I_1+I_j+I_n)}\n - 8 \\chi_1 \\chi_j f_2^{(I_1+2I_n)}  = \\mathfrak{D}_3 + \\mathfrak{D}_4.\n\\end{split}\n\\end{equation}\n\n\\medspace\n\nAs in (\\ref{new form}), we have\n\\begin{equation}\n\\label{new formbb} \\mathfrak{B}_1 \\ov{\\phi}(\\chi,\n0)^t=\\mathfrak{A}_1.\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathfrak{B}_1:= \\begin{pmatrix}\n{\\cal L}_j{\\cal L}_k \\phi_{hl} & {\\cal L}_j{\\cal L}_k \\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_k \\phi_{33} \\\\\n{\\cal L}_j{\\cal L}_\\beta \\phi_{hl} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{h\\alpha} & {\\cal L}_j{\\cal L}_\\beta\n\\phi_{33}\\\\\n{\\cal L}^2_1 {\\cal L}_j\\phi_{hl} & {\\cal L}^2_1 {\\cal L}_j\n\\phi_{h\\alpha} &\n{\\cal L}^2_1 {\\cal L}_j \\phi_{33} \\\\\n\\end{pmatrix}_{1\\leq j,h,l\\leq 2, {3\\leq \\alpha,\\beta\\leq n-1}} \\bigg|_{(0,0)}\n=\\begin{pmatrix}\nB_{1, jk}\\\\\nB_{1, j \\beta}\\\\\n\\mathfrak{B}_{1, 33}\n\\end{pmatrix}.\n\\end{equation}\nand $\\mathfrak{A}_1=(A_{1,11}, A_{1,12}, A_{1,22}, A_{1,1\\alpha},\nA_{1,2\\alpha}, \\mathfrak{A}_{1,33})^t$ where\n$\\mathfrak{A}_{1,33}=-{\\cal L}_1^2 {\\cal L}_j f_1|_{(0,0)} \\chi_1 -\n{\\cal L}_1^2 {\\cal L}_j f_2|_{(0,0)}$ $\\chi_2 =\n-(\\mathfrak{D}_1+\\mathfrak{D}_2) \\chi_1 - (\\mathfrak{D}_3 +\n\\mathfrak{D}_4) \\chi_2$, and $\\mathfrak{B}_{1,33}=({\\cal L}_1^2{\\cal\nL}_j  \\phi_{hl}|_{(0,0)},\\ {\\cal L}_1^2{\\cal L}_j\n\\phi_{h\\alpha}|_{(0,0)},\\ {\\cal L}_1^2{\\cal L}_j\n\\phi_{33}|_{(0,0)})$.\n\n\\medspace\n\nWe also have $\\mathcal{L}_1^2\\mathcal{L}_j\\phi|_{(0,0)} =\n2\\phi^{(2I_1+I_j)}+2i\\chi_j\\cdot\n2\\phi^{(2I_1+I_n)}+4i\\chi_1\\phi^{(I_1+I_j+I_n)} - 8\\chi_1\n\\chi_j\\cdot 2\\phi^{(I_1+2I_n)} - 4 \\chi_1^2 \\cdot\n2\\phi^{(I_j+2I_n)}$ so that\n\n\\begin{equation*}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{11}|_{(0,0)} = 2\\cdot 2i\n\\ov{e_{1,1j}} + 4i\\chi_j \\bigg(2i (\\frac{\\mu_{11}}{\\mu_1}\n\\ov{e_{1,11}} e_{1,11}\n+ \\frac{\\mu_{11}}{\\mu_2} \\ov{e_{2,11}} e_{2,11}) - \\frac{2i}{\\mu_{11}} f_1^{(I_1+2I_n)}\\bigg)\\\\\n& +4i\\chi_1 \\bigg(2i(\\frac{\\mu_{1j}}{\\mu_1} \\ov{e_{1,1j}}e_{1,11} +\n\\frac{\\mu_{1j}}{\\mu_2} \\ov{e_{2,1j}}e_{2,11})\\bigg)\n - 16 \\chi_1 \\chi_j\\bigg(-\\frac{2i}{\\mu_1} e_{1,11} f_1^{(I_1+2I_n)}\n- \\frac{2i}{\\mu_2} e_{2,11} f_2^{(I_1+2I_n)}\\bigg)\\\\\n& = -\\frac{2i}{\\mu_1}e_{1,11}(\\mathfrak{D}_1 + \\mathfrak{D}_2) -\n\\frac{2i}{\\mu_2}e_{2,11}(\\mathfrak{D}_3 + \\mathfrak{D}_4) -\n\\frac{1}{\\mu_{1j}\\chi_1} \\mathfrak{D}_1,\n\\end{split}\\end{equation*}\n\n\\begin{equation*}\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{12}|_{(0,0)} =  2 \\frac{2i\n\\sqrt{\\mu_1}}{\\sqrt{\\mu_1+\\mu_2}} \\ov{e_{2,1j}} +4i\\chi_j\n\\bigg(2i(\\frac{\\mu_{11}}{\\mu_1} \\ov{e_{1,11}} e_{1,12} +\n\\frac{\\mu_{11}}{\\mu_2}\n\\ov{e_{2,11}} e_{2,12}) -\\frac{2i}{\\mu_{12}} f_2^{(I_1+2I_n)} \\bigg) \\\\\n& + 4i\\chi_1 \\bigg(2i(\\frac{\\mu_{1j}}{\\mu_1} \\ov{e_{1,1j}} e_{1,12}\n+ \\frac{\\mu_{1j}}{\\mu_2} \\ov{e_{2,1j}} e_{2,12}) \\bigg)\n- 16\\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)} e_{1,12} - \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,12}\\bigg)\\\\\n& = -\\frac{2i}{{\\mu_1}}e_{1,12}(\\mathfrak{D}_1 + \\mathfrak{D}_2)\n-\\frac{2i}{\\mu_2}e_{2,12}(\\mathfrak{D}_3 + \\mathfrak{D}_4) -\n\\frac{1}{\\mu_{12} \\chi_1} \\mathfrak{D}_3,\n\\end{split}\\end{equation*}\n\n\n\\begin{equation*}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{22}|_{(0,0)} = 4i\\chi_j\n\\bigg(2i(\\frac{1}{\\mu_1} \\mu_{11} \\ov{e_{1,11}}e_{1,22}\n+ \\frac{1}{\\mu_2} \\mu_{11} \\ov{e_{2,11}}e_{2,22})\\bigg) \\\\\n& + 4i\\chi_1\\bigg(2i(\\frac{1}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}}e_{1,22}\n+ \\frac{1}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}}e_{2,22})\\bigg)\\\\\n& - 16\\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,22}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,22}\\bigg) \\\\\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n& = -\\frac{2i}{\\mu_1}e_{1,22}(\\mathfrak{D}_1 + \\mathfrak{D}_2) -\n\\frac{2i}{\\mu_2}e_{2,22}(\\mathfrak{D}_3 + \\mathfrak{D}_4),\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{1k}|_{(0,0)} = 2 \\cdot 2i\n\\delta_{jk}\\ov{e_{1,11}} + 4i \\chi_j \\bigg(2i(\\frac{1}{\\mu_1}\n\\mu_{11} \\ov{e_{1,11}}e_{1,1k}\n+ \\frac{1}{\\mu_2} \\mu_{11} \\ov{e_{2,11}} e_{2,1k}) \\bigg) \\\\\n& + 4i\\chi_1 \\bigg(2i(\\frac{1}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}}e_{1,1k}\n+ \\frac{1}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}} e_{2,1k})\n- \\frac{2i}{\\sqrt{\\mu_1}}\\delta_{jk} f_1^{(I_1+2I_n)}\\bigg)\\\\\n& - 16 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,1k}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,1k}\\bigg) \\\\\n& = - \\frac{2i}{\\mu_1}e_{1,1k} (\\mathfrak{D}_1 + \\mathfrak{D}_2) -\n\\frac{2i}{\\mu_2} e_{2,1k}(\\mathfrak{D}_3 + \\mathfrak{D}_4) -\n\\frac{1}{\\sqrt{\\mu_1} \\chi_j} \\delta_{jk}\\mathfrak{D}_2,\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{2k}|_{(0,0)} = 2 \\cdot \\frac{ 2i\n\\sqrt{\\mu_1}}{\\sqrt{\\mu_2}}\\delta_{jk}\\ov{e_{2,11}} + 4i \\chi_j\n\\bigg(2i(\\frac{1}{\\mu_1} \\mu_{11} \\ov{e_{1,11}}e_{1,2k}\n+ \\frac{1}{\\mu_2} \\mu_{11} \\ov{e_{2,11}} e_{2,2k}) \\bigg) \\\\\n& + 4i\\chi_1 \\bigg(2i(\\frac{1}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}}e_{1,2k}\n+ \\frac{1}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}} e_{2,2k})\n- \\frac{2i}{\\sqrt{\\mu_2}} \\delta_{jk}f_2^{(I_1+2I_n)}\\bigg)\\\\\n& - 16 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,2k}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,2k}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1}e_{1,2k} (\\mathfrak{D}_1 + \\mathfrak{D}_2) -\n\\frac{2i}{\\mu_2} e_{2,2k}(\\mathfrak{D}_3 + \\mathfrak{D}_4)\n - \\frac{1}{\\sqrt{\\mu_2} \\chi_j} \\delta_{jk}\\mathfrak{D}_4,\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n& \\mathcal{L}_1^2\\mathcal{L}_j\\phi_{33}|_{(0,0)} = 2 \\cdot\n\\frac{2\\mu_1}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}} \\ov{e_{2,1j}}\\\\\n& + 4i \\chi_j \\bigg(2i(\\frac{1}{\\mu_1} \\mu_{11}\n\\ov{e_{1,11}}e_{1,33} + \\frac{1}{\\mu_2} \\mu_{11} \\ov{e_{2,11}}\ne_{2,33}) - \\frac{2 \\mu_1}{\\sqrt{\\mu_1 \\mu_2(\\mu_1+\\mu_2)}}\nf_2^{(I_1+2I_n)}\n\\bigg) \\\\\n& + 4i\\chi_1 \\bigg(2i(\\frac{1}{\\mu_1} \\mu_{1j} \\ov{e_{1,1j}}e_{1,33}\n+ \\frac{1}{\\mu_2} \\mu_{1j} \\ov{e_{2,1j}} e_{2,33})\\bigg)\\\\\n& - 16 \\chi_1 \\chi_j \\bigg(-\\frac{2i}{\\mu_1} f_1^{(I_1+2I_n)}\ne_{1,33}\n- \\frac{2i}{\\mu_2} f_2^{(I_1+2I_n)} e_{2,33}\\bigg)\\\\\n& = - \\frac{2i}{\\mu_1}e_{1,33} (\\mathfrak{D}_1 + \\mathfrak{D}_2) -\n\\frac{2i}{\\mu_2} e_{2,33}(\\mathfrak{D}_3 + \\mathfrak{D}_4) + \\frac{i\n\\sqrt{\\mu_1}}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)} \\chi_1} \\mathfrak{D}_3.\n\\end{split}\n\\end{equation*}\n\nWe write $\\mathfrak{B}_2:=(Id - \\mathfrak{G}_1) \\mathfrak{B}_1$ and\n$\\mathfrak{A}_2:=(Id - \\mathfrak{G}_1) \\mathfrak{A}_1$ so that\n\\[\n\\mathfrak{B}_2 \\ov\\phi(\\chi, 0)^t = \\mathfrak{A}_2\n\\]\nwhere\n\\[\n\\mathfrak{G}_1:=\n\\begin{pmatrix}\n0 & 0 & 0 & \\frac{2\\chi_1}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_2}{\\chi_3} & 0 & \\frac{\\chi_1}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{2\\chi_2}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{\\chi_\\alpha}{\\chi_3} & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{\\mathfrak{D}_1+\\mathfrak{D}_2}{\\mu_1 \\chi_3} & 0\n&\n-\\frac{\\mathfrak{D}_3+\\mathfrak{D}_4}{\\mu_2 \\chi_3} & 0 & 0 \\\\\n\\end{pmatrix}.\n\\]\nBy the construction of $\\mathfrak{B}_2$ and $\\mathfrak{A}_2$, we see\nthat (\\ref{rel})(\\ref{e1}) and (\\ref{e2}) still hold. We further\ncalculate:\n\\begin{equation}\n\\begin{split}\n&\\mathfrak{B}_{2,33} = \\mathfrak{B}_{1,33} + \\frac{B_{1,13}}{\\mu_1\n\\chi_3} (\\mathfrak{D}_1 + \\mathfrak{D}_2) +  \\frac{B_{1,23}}{\\mu_2\n\\chi_3} (\\mathfrak{D}_3 + \\mathfrak{D}_4) = \\bigg(-\\frac{1}{\\mu_{1j}\n\\chi_1}, \\mathfrak{D}_1\\ - \\frac{1}{\\mu_{12}\n    \\chi_1} \\mathfrak{D}_3,\\ 0,\\\\\n&\\  - \\frac{1}{\\sqrt{\\mu_1} \\chi_3} \\mathfrak{D}_2 +\n\\frac{\\mathfrak{D}_1+\\mathfrak{D}_2}{\\sqrt{\\mu_1} \\chi_3}, 0, ...,\n0,\\ -\\frac{1}{\\sqrt{\\mu_2} \\chi_3} \\mathfrak{D}_4 +\n\\frac{\\mathfrak{D}_3+\\mathfrak{D}_4}{\\sqrt{\\mu_2} \\chi_3}, 0, ...,\n0,\\ \\frac{i \\sqrt{\\mu_1}}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)} \\chi_1}\n\\mathfrak{D}_3 \\bigg)\n\\end{split}\n\\end{equation}\nand\n\\begin{equation*}\\begin{split}\n& \\mathfrak{A}_{2,33} =\n-\\chi_1(\\mathfrak{D}_1+\\mathfrak{D}_2)-\\chi_2(\\mathfrak{D}_3+\\mathfrak{D}_4)\n+\\frac{\\mu_1\\chi_1\\chi_3}{\\mu_1\\chi_3}(\\mathfrak{D}_1+\\mathfrak{D}_2)\n+\\frac{\\mu_2\\chi_2\\chi_3}{\\mu_2\\chi_3}(\\mathfrak{D}_3+\\mathfrak{D}_4)\n= 0.\n\\end{split}\\end{equation*}\n\n\\medspace\n\nWe turn to $\\mathfrak{B}_{2,33} \\ov\\phi(\\chi, 0)^t =\n\\mathfrak{A}_{2,33}$ to have\n\\begin{equation}\\begin{split}\n&  -\\frac{\\mathfrak{D}_1}{\\sqrt{\\mu_1}\\chi_1} \\ov{\\phi_{1111}}\n-\\frac{\\mathfrak{D}_3}{\\mu_{12}\\chi_1}\\ov{\\phi_{12}} -\n\\bigg(\\frac{\\mathfrak{D}_2}{\\sqrt{\\mu_1}\\chi_3}\n- \\frac{\\mathfrak{D}_1+\\mathfrak{D}_2}{\\sqrt{\\mu_1} \\chi_3}\\bigg)\\ov{\\phi_{13}}\\\\\n&\\ \\ - \\bigg(\\frac{\\mathfrak{D}_4}{\\sqrt{\\mu_2}\\chi_3} -\n\\frac{\\mathfrak{D}_3+\\mathfrak{D}_4}{\\sqrt{\\mu_2} \\chi_3}\n\\bigg)\\ov{\\phi_{23}} + \\frac{i \\sqrt{\\mu_1}\n\\mathfrak{D}_3}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}\\chi_1}\\ov{\\phi_{33}}=0.\n\\end{split}\\end{equation}\nSubstituting (\\ref{rel}) to this equation, we obtain at $(\\chi, 0)$\n\\begin{equation*}\\begin{split}\n& -\\frac{\\mathfrak{D}_1}{\\sqrt{\\mu_1}\\chi_1}\n\\frac{\\chi_1}{\\chi_3}\\ov{\\phi_{13}}\n-\\frac{\\mathfrak{D}_3}{\\sqrt{\\mu_1+\\mu_2}\\chi_2}\n\\bigg(\\frac{\\sqrt{\\mu_1}}{\\sqrt{\\mu_1+\\mu_2}} \\frac{\\chi_2}{\\chi_3}\n\\ov{\\phi_{13}} + \\frac{\\sqrt{\\mu_2}}{\\sqrt{\\mu_1+\\mu_2}}\n\\frac{\\chi_1}{\\chi_3} \\ov{\\phi_{23}}\\bigg) \\\\\n& - \\bigg(\\frac{\\mathfrak{D}_2}{\\sqrt{\\mu_1}\\chi_3} -\n\\frac{\\mathfrak{D}_1+\\mathfrak{D}_2}{\\sqrt{\\mu_1} \\chi_3}\\bigg)\n\\ov{\\phi_{13}} - \\bigg(\\frac{\\mathfrak{D}_4}{\\sqrt{\\mu_2}\\chi_3} -\n\\frac{\\mathfrak{D}_3+\\mathfrak{D}_4}{\\sqrt{\\mu_2}\n\\chi_3}\\bigg)\\ov{\\phi_{23}} + \\frac{i \\sqrt{\\mu_1}\n\\mathfrak{D}_3}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)} \\chi_1} \\ov{\\phi_{33}} =\n0.\n\\end{split}\\end{equation*}\n\nA quick simplification gives at $(\\chi, 0)$\n\\begin{equation}\\begin{split}\n& - \\frac{\\mu_1 \\mathfrak{D}_3 }{\\sqrt{\\mu_1}(\\mu_1+\\mu_2)\n\\chi_3}\\ov{\\phi_{13}} + \\frac{\\mu_1\n\\mathfrak{D}_3}{\\sqrt{\\mu_2}(\\mu_1+\\mu_2)\\chi_3}\\ov{\\phi_{23}} +\n\\frac{i \\sqrt{\\mu_1} \\mathfrak{D}_3}{\\sqrt{\\mu_2(\\mu_1+\\mu_2)}\n\\chi_2}\\ov{\\phi_{33}} =0.\n\\end{split}\\end{equation}\nSince $\\phi^{(2I_1+I_j)}_{33}\\neq 0$ in Case D, the polynomial\n$\\mu_1 \\mathfrak{D}_3\\neq 0$ because of (\\ref{phi 1}). Hence we\nobtain the same formula (\\ref{33 eq}). As in the case A, we can\nfurther get $deg(F)\\le 3$. The proof for the case D$'$ is similar to\nthe case D.\n\\end{proof}\n\n\\medspace\n\nTherefore the proofs of Proposition \\ref{propdeg} and Theorem\n\\ref{mainthm} are complete.\n\n\n\n\n\n\\medskip\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Introduction}\n\n\nObservations of elliptical galaxies in clusters with {\\it XMM-Newton} \\ and {\\it Chandra} \\ have shown that the cluster environment has a profound impact on the evolution of galaxies. There are several mechanisms by which galaxies in clusters interact with both neighboring galaxies and the surrounding Intracluster Medium (ICM). Most galaxy-galaxy interaction mechanisms usually require low-velocity approaches to be efficient, and typically result in the stripping of both diffuse interstellar gas and a fraction of their stellar populations. Galaxy-ICM interactions commonly result in the stripping of the gas associated with the galaxy halo. In the most dramatic examples, ram pressure stripping can result in tails of both hot and cool gas trailing behind the galaxies for distances of $\\sim 100 {\\rm\\thinspace kpc}$ or longer.\nThe physical processes that operate within these tails, however, are not fully understood. While simulations have shown that simple analytic estimates are fairly accurate in predicting the radius of the gas disc that undergoes ram pressure stripping \\citep{Gunn1972}, complex physics is required to account for the morphology of the stripped gas tails and their multiwavelength properties \\citep[e.g.][]{Roediger2006,Roediger2007,Roediger2008,Tonnesen2010,Tonnesen2011,Tonnesen2012}. The viscosity of the ICM \\citep{Roediger2008b}, turbulence, and magnetic fields \\citep{Ruszkowski2012} are all expected to influence the morphology of the stripped gas and its evolution after it is deposited into the ICM. The relative contributions of these processes remain uncertain. \n\nArguably the ideal site to observe these interaction mechanisms is the nearby, bright Virgo Cluster. The most massive galaxies in this cluster have been studied in detail at multiple wavelengths, and X-ray observations have shown convincing evidence for profound interactions between the galaxies and the ICM. Long tails of cool, stripped gas have been observed trailing in the wake of Virgo ellipticals including NGC 4552 \\citep{Machacek2006}, and NGC 4472 \\citep{Kraft2011}. \n\nObservations of the Virgo Cluster have shown that it is composed of three separate sub-groups: one centered on M87, the Brightest Cluster Galaxy (BCG); one centered on M86, an elliptical galaxy roughly one degree ($\\sim 300 {\\rm\\thinspace kpc}$) to the west of M87; and the final sub-group centered on NGC 4472 (also known as M49) approximately $4.5^{\\circ}$ to the south of M87 \\citep{Binggeli1987,Binggeli1993,Bohringer1994}. Spectroscopic studies of M86 have shown that it has an absolute blue-shifted velocity of $\\sim 250 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$ while M87 has a red-shifted velocity of $\\sim 1300 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$. Given an average temperature of the ambient Virgo ICM of $\\sim 2.3 {\\rm\\thinspace keV}$ \\citep[][corresponding to a sound speed of $\\sim 700 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$]{Urban2011}, M86 must be traversing through the Virgo Cluster supersonically with a Mach number of at least $\\mathcal{M} \\gtrsim 2$. \n\nIn the vicinity of M86, additional galaxies are located within approximately $\\sim 100 {\\rm\\thinspace kpc}$ of one another in projection. The two most massive galaxies, M86 and M84, are separated by a projected distance of only $60 {\\rm\\thinspace kpc}$. Although these galaxies appear similar in optical imaging studies, further studies utilizing optical spectroscopy and X-ray observations show that they are interacting with the ambient Virgo ICM in different manners. Observations of M86 with {\\it Chandra} \\ and {\\it XMM-Newton} \\ \\citep{Finoguenov2004,Randall2008} show that it is undergoing significant ram pressure stripping, with diffuse X-ray emitting gas as well as cooler material trailing behind the galaxy for distances of approximately $\\sim 150 {\\rm\\thinspace kpc}$ in projection to the northwest. There is only sparse evidence for ram pressure stripping in observations of M84 with {\\it Chandra} \\ \\citep{Finoguenov2008}, but there is clear evidence for a powerful AGN outburst at the center of this galaxy \\citep{Finoguenov2008}. M84 has an absolute red-shifted velocity of $\\sim 1000 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$ \\citep{Smith2000,Trager2000}, approximately consistent with the measured velocity for M87. The region in between M84 and M86 is also claimed to be the site of bright intracluster light emission \\citep{Rudick2010}. Two other galaxies are detected in the X-rays nearby: NGC 4388 and NGC 4438. These spiral galaxies have absolute redshifted velocities of $2566 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$ \\citep[NGC 4388,][]{Rines2008} and $71 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$ \\citep[NGC 4438,][]{Kenney1995}, respectively.  \n\nIn this work we present, for the first time, an investigation into the thermodynamic structure spanning the entire region across these four galaxies. Utilizing the full information of five separate {\\it XMM-Newton} \\ observations, including new observations of M84, we are able to investigate the temperature and metallicity structure around these galaxies and investigate how interactions of these galaxies with each other and the ICM have transformed their gas reservoirs. \n\n\nThis paper is structured as follows: in Section 2 we discuss the data sets, their initial processing, and the production of images and spectral maps; in Section 3 we present our results; and in Section 4 we interpret their physical implications. We assume a nominal distance of 17 {\\rm\\thinspace Mpc} \\ to the Virgo Cluster, at which 1 arcsecond corresponds to 82 {\\rm\\thinspace pc}, and 1 arcminute to 4.92 {\\rm\\thinspace kpc}.\n\n \n\\begin{figure*}\n\\includegraphics[width=0.95\\textwidth]{BigMap_lr.ps}\n\\caption{\\label{Mosaic} Background-subtracted, exposure corrected mosaic of five {\\it XMM-Newton} \\ pointings surrounding M86 (in units of $\\rm{cts} {\\rm\\thinspace s}^{-1}$), utilizing the MOS+PN detectors in the energy band of $0.3-1.0 {\\rm\\thinspace keV}$. The four X-ray brightest Virgo Cluster members observed in this mosaic (M86, M84, NGC4388, and NGC 4438) are labeled. Diffuse gas is observed trailing to the northwest of M86, to the east of M86 in the direction of NGC 4438, to the south of M84, and to the east of NGC 4388.  }\n\\end{figure*}\n\n\n\\section{Data Processing and Analysis Methods}\\label{datared}\nA total of five separate {\\it XMM-Newton} \\ pointings were analyzed, the details of which are listed in Table \\ref{Observations}. The data products were cleaned and processed using the {\\it XMM-Newton} \\ Science Analysis System (SAS) (version 11.0), and in particular the Extended Source Analysis Software \\citep[{\\it ESAS},][]{Snowden2008} package included with SAS. We have utilized events from the two MOS and PN detectors for this study. Calibration products and event lists were first produced using the standard SAS routines. Good time intervals (GTI's) for each event list were determined using the {\\small MOS-FILTER} and {\\small PN-FILTER} routines. Four of the five pointings had little time contaminated by flares, but the most recent observation (of M84, Observation \\# 0673310101, PI S. Ehlert) was heavily contaminated by flaring, with only $\\sim 40$ ks of the original 130 ks MOS exposure deemed suitable for analysis. No further evidence for flaring was observed in any of the event lists after GTI filtering. \n\\begin{table}\n\\caption{\\label{Observations}Summary of the five {\\it XMM-Newton} \\ observations utilized in this study. Exposure times are the net exposure after all cleaning and processing as described in Section \\ref{datared} for each detector. The M84 observation (Obs \\# 0673310101, denoted by an asterisk) was heavily contaminated by flares. The original event list was split into two separate event lists for each MOS instrument, the summed exposure time of which is shown below.     }\n\\centering\n\\begin{tabular}{ c  c c c c} \n\\\\ \\hline {Obs \\#} & {Obs. Date} & MOS1 (ks) & MOS2 (ks) & PN (ks) \\\\ \\hline\\hline \n\n0108260201 & July 01 2002 & 62.44 & 63.53 & 43.68\\\\\n0110930701 & December 12 2002 & 8.46 & 10.05  & 3.84\\\\ \n0210270101 & December 19 2004 & 25.42 & 25.25 & 23.79\\\\ \n0210270201 & December 27 2004 & 20.38 & 19.92 & 19.02\\\\ \n0673310101* & June 01 2011 & 37.14 & 42.96 & 15.42\\\\ \n\n\n\\hline\n\\end{tabular}\n\\end{table}\n\n \n\\subsection{Imaging Analysis}\nBackground-subtracted, exposure corrected images of all five pointings were produced in the soft energy band, $0.3-1.0 {\\rm\\thinspace keV}$. The backgrounds for imaging analysis were determined from customized blank-sky observations \\citep{Carter2007} and closed-filter wheel observations to account for the instrumental background. We use blank sky files with similar pointing directions and Galactic absorption column densities to M84 and M86. The closed-filter wheel counts images were subtracted from both the blank-sky and science exposures after renormalizing to match the total counts in each of these images outside of the field-of-view. The remaining blank sky emission was then subtracted from the science images. We also accounted for out-of-time (OOT) events in the PN detector images for each pointing using standard procedures. \n\n\\begin{figure*}\n\\includegraphics[width=0.95\\textwidth]{Sloan_withXRay_lr.ps}\n\\caption{\\label{Sloan} An r-band image of the region surrounding M86 from the Sloan Digital Sky Survey. The X-ray surface brightness contours are overlaid in white, which show the stripped X-ray emitting gas trailing these galaxies. The stripped gas tail is especially pronounced for the M86 at the center of the image. Tails of stripped gas are observed to the south of M84 and to the northwest of M86. Diffuse soft emission is detected trailing eastward from M86 in the direction of NGC 4438. Faint, diffuse emission is also detected at the position of the edge-on spiral galaxy NGC 4402, located $\\sim 10 \\arcmin$ to the north of M86.   }\n\\end{figure*}\n\n\n\n\n\\subsection{Thermodynamic Mapping}\nOur {\\it XMM-Newton} \\ observations are sufficiently deep for high signal-to-noise spectra to be extracted from relatively small regions. This enables us to carry out detailed, spatially resolved measurements of the thermodynamic quantities across the entire mosaic. All spectral analysis was carried out using {\\small XSPEC} \\rm  \\citep[][version 12.6]{Arnaud2004}. Instrumental background was determined for the source spectra using closed filter wheel observations of the detectors renormalized to have the same count rate in the $10-12 {\\rm\\thinspace keV}$ energy band. We subtracted the out-of-time events from the PN spectra before performing these fits. We modeled the sky background for each spectral bin using the background model of \\cite{Urban2011}, renormalized to account for the appropriate region geometries. This model assumes a sum of three components: an absorbed power law from the unresolved point sources \\citep{Deluca2004}, absorbed thermal emission with $kT \\sim 0.2 {\\rm\\thinspace keV}$ from the Galactic halo \\citep{Kuntz2000} and the unabsorbed thermal emission from the Local Hot Bubble \\citep{Sidher1996,Kuntz2000}. Tests show that no additional background components are necessary.  Allowing the temperatures, photon indexes, or normalizations of the background components to vary did not change the results in a significant manner.   \n\n\n\n\\subsubsection{Regions of Interest}\\label{ROI}\nWe identified the independent regions of interest for our spectral analysis by binning the background subtracted image into regions of equal signal to noise. Here we made use of the Weighted Voronoi Tessellations binning algorithm by \\cite{Diehl2006}, which is a generalization of the Voronoi binning algorithm of \\cite{Cappellari2003}. We binned the image using two signal-to-noise thresholds (hereafter the low S\/N and high S\/N maps, respectively). Spectra were extracted from each spectral region, and appropriate response files were generated using the {\\small RMFGEN} and {\\small ARFGEN} tools. All spectral fits were performed in the energy band $0.4-7.0 {\\rm\\thinspace keV}$, although we ignored the energy range $1.2-1.8 {\\rm\\thinspace keV}$ in order to avoid the instrumental Al and Si emission lines in the MOS detectors. In the resulting energy band, there are roughly $\\sim 4000$ counts per region in our low S\/N maps and $\\sim 7000$ counts per region for our high S\/N maps.\n\n\n\\subsubsection{Spectral Modeling and Thermodynamic Quantities}\n\nFor the low S\/N regions, we fit the spectra with an absorbed, single temperature {\\small APEC} plasma model. We utilized the photoelectric absorption model of \\cite{McCammon1992} and fixed the Galactic absorption column density to the value derived by \\cite{Kalberla2005} $\\mysub{N}{H}=2.13 \\times 10^{20} {\\rm\\thinspace cm}^{-2}$. We assume the solar element abundance ratios given by \\cite{Grevesse1998}. The temperature, metallicity, and the normalization were all independent free parameters. Best fit model parameters were obtained by minimizing the modified C-statistic in {\\small XSPEC}. All uncertainties quoted are $68\\%$ ($\\Delta C=1$) confidence intervals. \n\nThe high S\/N spectra were fitted with a similar model, but with an additional thermal component to account for the ambient Virgo Cluster emission viewed in projection. The temperature of this thermal emission was fixed to  2.3 {\\rm\\thinspace keV}, in agreement with measurements of the diffuse Virgo ICM at the same distance from M87 \\citep{Urban2011}. The metallicities of the two thermal components were linked to be the same. The exact choice of the temperature of the ambient Virgo ICM does not have a strong influence on the measured temperature or metallicity for the second component.   \n\nTo search for multi-temperature structure in the high S\/N map, we also attempted to fit the spectra with a two-temperature plus cooling flow model ({\\small APEC+APEC+MKCFLOW}). For the cooling flow component we set the upper and lower temperatures to $1 {\\rm\\thinspace keV}$ and $0.1 {\\rm\\thinspace keV}$, respectively. All of our measured temperatures have an uncertainty of less than $\\sim 10\\%$, regardless of the model.   \n\n\n\\section{Results}\nThe background subtracted, exposure corrected image of the region surrounding M86 is shown in Figure \\ref{Mosaic}. To the north-west of M86 a long tail of diffuse gas is observed, which has been discussed in detail by \\cite{Finoguenov2004} and \\cite{Randall2008}. The length of this tail is $\\sim 150 {\\rm\\thinspace kpc}$ in projection, although given the large line-of-sight velocity of M86 with respect to the ambient Virgo ICM this is likely a conservative lower limit to its true length. The width of this tail in projection, however, is roughly uniform along its entire length. To the east of M86, diffuse emission is observed out to the neighboring galaxy NGC 4438. Diffuse gas is also detected for a distance of approximately $\\sim 15 {\\rm\\thinspace kpc}$ to the south of M84. These tails of X-ray emission trailing M86 and M84 extend beyond the bulk of the optical galaxy emission, as shown in the Sloan Digital Sky Survey image of this field (Figure \\ref{Sloan}). There is no evidence for diffuse emission that bridges between M84 and M86. There is also tentative evidence for diffuse X-ray emission trailing to the east of NGC 4388.  \n\n\n\n\\subsection{Temperature Structure}\nMaps of the temperature and metallicity structure across the region are shown in Figure \\ref{Thermo}. Cool gas with a temperature of $\\sim 1 {\\rm\\thinspace keV}$ is detected in various directions around M86, with the the majority of this gas phase located in the regions immediately surrounding the galaxy and to the north-west of the galaxy. This $\\sim 1 {\\rm\\thinspace keV}$ component traces the morphology of M86's X-ray tail, and originates from the ram pressure stripping of M86 \\citep[e.g.][]{Forman1979,Fabian1980,Nulsen1982,White1991,Finoguenov2004,Randall2008}. \n\nTo the east of M86 is a site of cooler X-ray emitting gas, observed to have a temperature of $\\sim 0.6 {\\rm\\thinspace keV}$, spatially coincident with the brighter diffuse X-ray emission spanning between M86 and NGC 4438. Gas with a temperature of $\\sim 0.6 {\\rm\\thinspace keV}$ is also observed just to the north of M86, extending to the northwest in the same direction as the $\\sim 1 {\\rm\\thinspace keV}$ component. Fitting each spectral bin in the region between M86 and NGC 4438 with a two-temperature plus cooling flow model results in no detection of significant cooling flows. For any particular bin, these fits constrain the spectroscopic cooling rate of X-ray gas from $1.0 {\\rm\\thinspace keV}$ to $0.1 {\\rm\\thinspace keV}$ to $\\lesssim 0.01 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$, and the upper limit for the integrated spectroscopic cooling rate across all of the bins between these two galaxies is $\\lesssim 0.1 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$.  \n\nIn the vicinity of M84, we detect the presence of cool ($\\sim 0.7 {\\rm\\thinspace keV}$) gas trailing to the south of the galaxy, out to a distance of $\\sim 15 {\\rm\\thinspace kpc}$. This is significantly larger than the scale of AGN feedback in this system (see Figure \\ref{M84Zoom}). The temperature of this gas is observed to increase smoothly to the south, until it becomes indistinguishable from the ambient ICM. The presence of cool gas trailing to the south of M84 suggests ram pressure stripping of its X-ray halo amidst northward motion. \n\nWe use the presence of this tail to provide an order-of-magnitude estimate of the rate at which X-ray gas is stripped from M84. In order to determine the gas mass in the tail, we assume it has a prolate ellipsoidal geometry with a semi-major axis of $7.5 {\\rm\\thinspace kpc}$ and a semi-minor axis of $4.2 {\\rm\\thinspace kpc}$ (shown in Figures \\ref{M84Zoom}). This geometry is based on the extent to which we can detect cooler ($\\sim 1 {\\rm\\thinspace keV}$) gas to the south of the cavity regions. We only include the region to the south of the AGN-driven cavities seen in {\\it Chandra} \\ observations of M84. Based on the surface brightness enhancement in this region, we estimate that the density of the gas in the tail is $\\sim 0.005 {\\rm\\thinspace cm}^{-3}$, roughly a factor of 2 denser than the surrounding Virgo ICM \\citep{Urban2011}. With these assumptions, we determine that a mass of $\\sim 7 \\times 10^{7} \\mysub{M}{\\odot}$ of X-ray gas has been stripped by the bulk motion of M84 through the Virgo ICM. Using the sound crossing time as an estimate for the time scale over which this stripping occurred and an assumed sound speed of $700 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$, the overall mass stripping rate is calculated to be $\\sim 7 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$. Since there is no evidence for jumps in the ICM temperature or surface brightness indicative of a shock front to the north of M84, this galaxy is likely traversing through the ICM at sub-sonic speeds\\footnote{Since M84 and M87 have similar line-of-sight velocities, it is also unlikely that M84 has a significant velocity component through the Virgo ICM along the line-of-sight. This maximizes the likelihood that such a shock front, if present, would be detected.}. Resultingly, the mass stripping rate we estimate is likely an upper limit as to the true rate at which M84's X-ray halo gas is being stripped by ram pressure.   \n\n\nTo the west of M86 in the direction of M84, a clear temperature gradient is observed over the entire region. Between the two galaxies is a $\\sim 20 {\\rm\\thinspace kpc}$ band of emission consistent with the ambient Virgo ICM, in both one-temperature and two-temperature fits.\n\n\n\\subsection{Metallicity Structure}\n\nAlthough the single-temperature and two-temperature fits qualitatively agree in their metallicity measurements (see Figures \\ref{Thermo}b and \\ref{Thermo}d), we focus our discussion on the measurements from the two-temperature fits. These measurements are performed at higher signal-to-noise and are less sensitive to biases in metallicity measurements that arise by modeling multi-phase emission with a single temperature \\citep{Buote2002}. For comparative purposes, the metallicity of the Virgo cluster at the same radius from M87 \\citep{Urban2011} is approximately $Z \\sim 0.3 \\ \\mysub{Z}{\\odot}$. \n\nThe metal-richest gas is located near the center of M86, which has a measured metallicity of at least $\\sim 1 \\mysub{Z}{\\odot}$. Metal-rich gas ($\\gtrsim 0.5 \\mysub{Z}{\\odot}$) is observed in the direction of M86's stripped tail, out to distances of $\\sim 90-100 {\\rm\\thinspace kpc}$. There is no strong evidence for metal-rich gas spanning between M86 and NGC 4438. However, it is possible that a metallicity-measurement bias arising from unaccounted temperature phases is still present here. A smaller enrichment in metals ($Z \\gtrsim 0.5 \\mysub{Z}{\\odot}$) is observed near the core of M84.  \n\n\n\\begin{figure*}\n\\includegraphics[width=0.47\\hsize]{MosaicTemp1_fix_lr.ps}\n\\includegraphics[width=0.47\\hsize]{MosaicZ1_fix_lr.ps}\\vspace{0.5mm}\n\\includegraphics[width=0.47\\hsize]{MosaicTemp2_fix_lr.ps}\n\\includegraphics[width=0.47\\hsize]{MosaicZ2_fix_lr.ps}\n\n\n\n\\caption{\\label{Thermo} Temperature and metallicity maps of the $\\sim 1^{\\circ}$ region surrounding M86, including the galaxies M84, NGC 4388, and NGC 4438. The upper row shows our low S\/N ($\\sim 4000$ counts per bin) temperature and metallicity maps. The values show our best-fit parameters using a single-temperature thermal emission model. The lower row shows temperature and metallicity maps from our high S\/N spectra ($\\sim 7000$ counts per bin). The values in these maps are derived from a two-temperature fit, with the second temperature fixed to $2.3 {\\rm\\thinspace keV}$ and the metallicities of both components tied to the same value. This two-temperature fit optimally identifies regions hosting gas significantly cooler than the ambient Virgo Cluster gas, which has an assumed temperature of $2.3 {\\rm\\thinspace keV}$. The inclusion of an additional temperature component also reduces metallicity measurement biases.}\n\n\\end{figure*}\n\n\n\\subsection{Ultraviolet Observations with {\\it GALEX}}\n\nWe have also utilized publicly available ultraviolet (UV) observations of the region between M86 and NGC 4438 with {\\it GALEX} in order to search for star formation in this region. We used observation ID ``NGA\\_Virgo\\_MOS10'' (observation date 2004-03-11), part of the {\\it GALEX} Ultraviolet Virgo Cluster Survey \\citep{Boselli2011}, which is well centered between the two galaxies. There are no siginficant detections of young stellar populations in the far UV image of this field. Converting the observed surface brightness of far-UV photons into a star formation rate using \\cite{Kennicutt1998}, we place a strict upper limit on the star formation rate in the region of the H$\\alpha$ filaments of $\\lesssim 0.1 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$. \n\n\n\n\\section{Discussion}\n\nUsing a mosaic of {\\it XMM-Newton} \\ observations of the region surrounding M86, we have identified multiple sites of stripped X-ray emitting gas associated with all four galaxies: M86, M84, and NGC 4438, and NGC 4388. Although the evidence for stripped X-ray emitting gas near NGC 4388 is weak, radio observations of this galaxy show a tail of \\ion{H}{i} gas trailing to the north of this galaxy, with a velocity structure consistent with NGC 4388 \\citep[e.g.][]{Oosterloo2005}. All four of these galaxies are therefore undergoing some form of ram pressure stripping, although each galaxy has its own unique observational signatures associated with it. The stripped X-ray emitting gas trailing to the northwest of M86 and the \\ion{H}{i} tail trailing to the north of NGC 4388 have been detected and discussed in previous work \\citep[e.g.][]{Randall2008,Oosterloo2005}. The stripped X-ray emitting gas to the south of M84 and the cool X-ray emitting gas stretching from M86 to NGC 4438 have not been discussed at length\\footnote{The diffuse tail of gas trailing to the south of M84 was discussed briefly in \\cite{Randall2008}, although there were insufficient counts in the {\\it Chandra} \\ data to resolve its thermodynamic structure.}. We expand our discussion of these two sites of ram pressure stripping below.\n\n\n\\begin{figure*}\n\\includegraphics[width=0.47\\hsize]{M84_SBTail_lr.ps}\n\\includegraphics[width=0.47\\hsize]{M84_TTail_fix_lr.ps}\n\\caption{\\label{M84Zoom} A zoomed in view of M84 with {\\it XMM-Newton}. Both of these images show the same field of view. The dashed ellipse denotes the region which we use to estimate the rate at which X-ray halo gas is being stripped from M84.  {\\it Left:} The background subtracted, exposure corrected image of M84 in the $0.3-1.0 {\\rm\\thinspace keV}$ energy band. The black contours are derived from {\\it Chandra} \\ observations of this galaxy, which show the extent of the AGN-driven cavities \\citep{Finoguenov2008}. {\\it Right:} The corresponding temperature map of this same region, in units of {\\rm\\thinspace keV}. The bin map is similar to that described in  \\S \\ref{ROI}, although with a only $\\sim 250$ net counts per bin in order to resolve smaller scale features. Each bin is fit with a single-temperature plasma model with Galactic absorption (\\small{PHABS $\\times$ APEC}). The same {\\it Chandra} \\ contours are overlaid in black. The brighter gas trailing to the south of M84 is shown to be significantly cooler than the ambient ICM.}  \n\\end{figure*}\n\n\nWhile all four of the galaxies lie near one another in projection, the X-ray data combined with absolute velocity measurements along the line-of-sight suggest that only NGC 4438 and M86 are interacting with one another in any substantial way. There is no evidence in either the X-ray imaging or spectral maps for diffuse emission or cool gas bridging between any other pair of galaxies in a manner suggesting galaxy-galaxy interactions. The reported intracluster light (ICL) between M86 and M84 \\citep{Rudick2010} therefore appears likely to be a projection effect.  \n\n\\subsection{Correlation of the Coolest X-ray Gas and H$\\alpha$ Filaments}\n\n\nIn Figure \\ref{WithHA}a, we show the temperature map of Figure \\ref{Thermo}c zoomed in to better show the $\\sim 0.6 {\\rm\\thinspace keV}$ gas plume to the east of M86. We have overlaid the H$\\alpha$ filaments identified by \\cite{Kenney2008}, which clearly show that the H$\\alpha$ filaments are located at the same positions as the lowest temperature phases of X-ray emitting gas. This region, at least the half closer to M86, is also observed to be a site of HI emission \\citep{Oosterloo2005}, although based on the velocity structure of that emission it may be primarily associated with NGC 4388 to the south of M86. On the other hand, some of the \\ion{H}{i} filaments detected near M86 have velocity structure similar to M86 \\citep{Kenney2008}. \n\nA similar spatial coincidence between cool X-ray emitting gas and H$\\alpha$ filaments has been seen in a stripped tail behind the galaxy ESO 137-001 falling into the cluster Abell 3627 \\citep{Sun2007}. The stripped gas trailing behind ESO 137-001 is a site of {\\it in situ} star formation taking place outside of the galaxy within the ICM. A similar coincidence of cool X-ray emitting gas (at temperatures of $\\sim 0.6 {\\rm\\thinspace keV}$) with filaments of H$\\alpha$ emission (at temperatures of $\\sim 10^{4} {\\rm\\thinspace K}$) has also been observed around the BCGs of nearby cool core clusters such as the Perseus Cluster \\citep{Sanders2007,Fabian2008,Fabian2011}, Centaurus Cluster \\citep{Sanders2002,Sanders2008}, and M87 \\citep[][]{Werner2010,Werner2012}. \n\n\n\n\n\\subsection{On The Origin of the H$\\alpha$ Filaments}\n\n\nThe origin of this $\\sim 10^{4} {\\rm\\thinspace K}$ phase of gas spatially coincident with soft X-ray emission between M86 and NGC 4438 is likely a previous collision between the two. The existing multiwavelength data are all consistent with this scenario, although further data will be required to rigorously confirm this. The X-ray emitting gas in this filament region appears poorer in metals than the gas detected at the center of M86, implying that this cool X-ray emitting gas likely originated in NGC 4438. Measurement biases, due to the presence of multiple temperature phases unaccounted for in our models cannot be ruled out as the reason for the observed low metallicity in this region, however.\n\n\nIn the absence of sources of heating, this stripped gas may be expected to cool to star-forming temperatures, as is observed in the galaxy  ESO 137-001 in Abell 3627 \\citep{Sun2007}. Uncertainties in the filament geometry (specifically the depth of the filament regions along the line of sight) limit our ability to determine the cooling time of the cool gas unambiguously. Based on the combined analysis of all of the spectral bins in this filament region discussed above, we can place an upper limit on the integrated cooling rate from $1 {\\rm\\thinspace keV}$ down to $0.1 {\\rm\\thinspace keV}$ across the whole region of $\\lesssim 0.1 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$. This is in agreement with the upper limit on star formation imposed by far UV emission of this same region discussed above. \n\nSpiral galaxies such as NGC 4438 typically have X-ray halos with an observed temperature of $\\sim 0.2 {\\rm\\thinspace keV}$, while the gas associated with the group halo of M86 has a temperature of $\\sim 1 {\\rm\\thinspace keV}$. The presence of $\\sim 0.6-0.7 {\\rm\\thinspace keV}$ gas in between the two galaxies suggests a possible mixture of gas phases. This ``bridge'' of cool gas between the two galaxies may arise either from ram pressure stripping of NGC 4438 by the surrounding Virgo ICM, or from the recent encounter between NGC 4438 and M86. The stripping medium in this latter scenario could either be the X-ray gas halo of M86 or the intragroup medium surrounding it, both of which could result in the observed filament of $\\sim 0.6-0.7 {\\rm\\thinspace keV}$ gas. \n\n\nThe spatial coincidence between H$\\alpha$ emitting filaments with $\\sim 0.6 {\\rm\\thinspace keV}$ X-ray emitting gas is similar to that observed in the centers of cool core clusters undergoing AGN feedback. It is therefore possible that similar physical processes drive the evolution of these distinct filament regions, at least after an initial disturbance. As discussed in \\cite[e.g.][]{Fabian2008,Fabian2011,Werner2012}, a potential energy source powering the FIR through UV emission (including the H$\\alpha$+\\ion{N}{ii} optical line emission) in cooling core clusters are hot ICM particles penetrating into the filaments of cold gas, resulting in collisional ionization and heating \\cite[see][]{Ferland2008,Ferland2009}. \\cite{Werner2012} propose that shear instabilities due to shocks or the motion of the filaments through the ambient ICM may facilitate mixing between the hot and cold gas phases. The physics of the multiphase gas in the M86 filament region, which is exposed to shear instabilities as it moves through the Virgo ICM and merger shocks due to the infall of the M86 group, may therefore share similarities with the physics of the optical emission line nebulae seen in the cooling cores of galaxy clusters. One key difference between the filament region discussed here with those observed near cool core clusters, however, is the nature of the initial disturbance. Rather than instabilities triggered by AGN feedback, the collision between M86 and NGC 4438 would be the source of the shearing here. Such triggering may be further encouraged by M86's supersonic motion and subsequent shock heating. \n\nSites within $\\sim 10 {\\rm\\thinspace kpc}$ of the two galaxies have recently been shown to host CO($1 \\rightarrow 0$) line emission, suggesting that molecular hydrogen gas is present between the two galaxies for time scales of at least $\\sim 100 $ Myr \\citep{Dasyra2012}. This molecular gas is inferred to originate primarily from NGC 4438, similar to what we infer for the X-ray emission. Further observations that place stricter constraints on the star formation rate and detailed spectral properties of this region, as well as the mass of cold gas present from emission lines such as \\ion{C}{ii}, will be required to test the connection between the H$\\alpha$ emission presented here and that observed in cool core clusters.  \n\n\n\n\n\\subsection{Combining AGN Feedback with Ram Pressure Stripping}\n\nThe detection of a clear tail of X-ray emission trailing to the south of M84 offers another example of a galaxy where feedback and ram pressure stripping work in conjunction to strip its X-ray halo. Ram pressure stripping combined with AGN feedback are also observed in Sersic 159-03 \\citep{Werner2011} and the Virgo elliptical galaxy NGC 4552 \\citep{Machacek2006,Machacek2006b}, and similar processes are observed at larger scales in the Ophiuchus Cluster \\citep{Million2010b} and MACSJ1931.8-2634 \\citep{Ehlert2011}. In these systems, the combined influences of these two processes can rapidly suppress or even destroy cool cores associated with the central regions of the galaxy\/galaxy cluster. It appears as though a configuration where both AGN feedback and ram pressure stripping occur simultaneously is more common than initially expected.  \n\nThe energy injected into the X-ray halo from AGN feedback may allow M84 to be stripped more efficiently now than in its past. At its current stripping rate of $\\lesssim 7  \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$ estimated above, all of the X-ray gas should be stripped from this galaxy within $\\sim 200-300 $ Myr, a time scale likely far shorter than the time elapsed since M84 first crossed the Virgo Cluster's virial radius. We determine whether losses from stellar winds may be sufficient to replenish the X-ray halo of M84, assuming a single-age passively evolving stellar population with a Salpeter IMF. We use the stellar mass-loss rate from \\cite{Ciotti1991} to estimate this rate, $\\mysub{M}{\\star}(t)$,  as\n\n\\begin{equation}\n\\mysub{M}{\\star}(t) \\simeq  1.5 \\times 10^{-11}\\mysub{L}{B}\\mysub{t}{15}^{-1.3}\n\\end{equation}\n\n\\noindent where \\mysub{L}{B} is the present-day B-band luminosity in units of \\mysub{L}{B,\\odot}\n\\citep[$\\mysub{L}{B} = 4.5 \\times 10^{10} \\mysub{L}{B,\\odot}$ for M84;][]{Finoguenov2002} and \\mysub{t}{15} is the age of the stellar population in units of 15 Gyr\\footnote{This formula is valid in the range from $\u223c0.5$ to over $15$ Gyr. }. The local stellar mass-loss rate, assuming that the current stellar population is $10$ Gyr old, is of the same order-of-magnitude as the rate of ram pressure stripping ($\\mysub{M}{\\star} \\sim 1 \\mysub{M}{\\odot} {\\rm\\thinspace yr}^{-1}$).   \n\n\n\\subsection{On the Frequency of Stripping in Cluster Member Galaxies}\nPerhaps surprisingly, each of the four X-ray emitting galaxies detected in this mosaic show evidence of being stripped as they traverse through the ICM. Although these galaxies share some common features, no two galaxies are observed to have stripped gas tails sharing all of the same properties. The prevalence of gas trailing in the wake of galaxies in this busy region of the Virgo Cluster suggests that stripping processes may play an important role for nearly all galaxies in clusters. The particular observational signatures associated with the stripping process may depend strongly on the morphology, mass, and gas content of the host galaxy, but it appears typical that gas initially hosted by galaxies is efficiently stripped by the ICM. \n\n\nFor more massive elliptical galaxies hosting large masses of X-ray emitting gas like M86 (likely the central galaxy of an infalling group) and M84, the stripping is most readily observed as tails of X-ray emitting gas trailing the galaxies. The stripped X-ray gas behind these two galaxies, however, suggest that the astrophysical processes that drive the stripping may vary between different sites. One key difference between the stripped X-ray gas between M86 and M84 is the relative length of the apparent tails, which may indicate differences in the gas physics between the two sites. The tail behind M86 remains roughly uniform in width and coherent for distances of at least $\\sim 150 {\\rm\\thinspace kpc}$ in projection without significant evidence for mixing with the surrounding ICM. The true length of this tail is also likely significantly larger than its observed length in projection due to M86's high velocity with respect to Virgo ($\\sim 1500 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$) along the line-of-sight. Maintaining such cohesion over these distances suggests that the stripped gas is undergoing a laminar flow even as the galaxy traverses through the ICM supersonically, and subsequently suggests that the stripped gas may have high viscosity. The gas trailing behind M84, on the other hand, is only observable for distances of $\\sim 15 {\\rm\\thinspace kpc}$ in surface brightness and temperature maps, and the lower velocity of M84 with respect to the Virgo ICM along the line-of-sight ($\\sim 200-300 {\\rm\\thinspace km} {\\rm\\thinspace s}^{-1}$) indicates that this tail is probably not significantly longer than its projected length. The temperature structure in this tail is also observed to increase smoothly with distance from M84. These may be evidence for more efficient mixing between the stripped gas and the surrounding ICM at this site, perhaps associated with a more turbulent flow of the stripped gas. Deeper observations will be required to determine the full length of the stripped gas trailing M84 and subsequently the extent to which turbulence and magnetic fields may be necessary to account for the thermodynamic structure in M84's tail. \n\nSpiral galaxies contain significant masses of gas at much lower temperatures ($\\sim 10-100 {\\rm\\thinspace K}$). The observed differences in the stripped cold gas behind NGC 4388 and NGC 4438 suggest that the manner in which these phases interact and intersperse with the ICM may vary also significantly from site to site, and that the presence of shear instabilities or shocks may play an important role in these processes. This also appears to be true for the X-ray halos, especially when the disruptions due to the surrounding environment are more extreme. Deeper multiwavelength observations will again be necessary to fully understand the range of gas temperatures present in these stripped tails and test the stripping processes from start to finish. \n \n\n\n\\begin{figure*}\n\\includegraphics[width=0.47\\hsize]{Temp_WithHA_fix_lr.ps}\n\\includegraphics[width=0.47\\hsize]{Xray_WithHA_lr.ps}\n\\caption{\\label{WithHA} The correlation between the cool X-ray gas and H$\\alpha$ emission detected by Kenney et al. 2008. (a) Temperature map of Figure \\ref{Thermo}d zoomed in to emphasize the temperature structure surrounding M86 and overlaid with contours of H$\\alpha$ imaging. There is a clear correlation of the H$\\alpha$ emission with the coolest gas phases in both the eastern and northern plumes extending out from M86. The centers of M86 and NGC 4438 are denoted by the cyan crosses. (b) The X-ray surface brightness image of Figure \\ref{Mosaic}, zoomed in to the same region as (a). The H$\\alpha$ image contours are overlaid in cyan. }\n\\end{figure*}\n\n\\section{Conclusions}\nWe have measured the X-ray spectral properties in a contiguous $\\sim 1^{\\circ} \\times 1^{\\circ}$ region spanning four Virgo Cluster galaxies, allowing us to view interactions between them. Our key results are as follows: \n\n1) There is clear evidence for cool ($\\sim 0.6 {\\rm\\thinspace keV}$) gas trailing to the northwest of M86 as reported in previous studies, to the east of M86 in the direction of NGC 4438, and to the south of M84. M86 appears to be surrounded in a halo of $\\sim 1 {\\rm\\thinspace keV}$ gas, which we interpret as the intragroup medium of a recently merged subcluster undergoing ram pressure stripping by the ambient Virgo ICM. We also observe tentative evidence for stripped X-ray gas to the east of NGC 4388. \n\n2) The presence of stripped gas associated with all four of the largest galaxies in this field of view indicates that, while the particular astrophysical details may vary from site to site, ram pressure stripping is a common occurrence in cluster member galaxies, and has important implications for the heating and enrichment of the ICM. \n\n3) Filaments of H$\\alpha$ emission surrounding M86 are observed at locations spatially coincident with enhanced X-ray emission at temperatures of $\\sim 0.6 {\\rm\\thinspace keV}$. These filament regions have similar X-ray spectra to H$\\alpha$ filaments observed in the centers of cool core clusters. One key difference between this filament region and those detected in cool core clusters, however, is the likely source of the initial disturbance that triggers their formation.  \n\n4) AGN feedback and ram pressure are both observed in M84, and  in tandem these two processes may be encouraging particularly efficient stripping. Without replenishment, M84's X-ray halo gas would be depleted on time scales of $\\sim 300$ Myr. Mass loss from stellar winds, however, appears to be sufficient to replenish the M84's X-ray halo, at least in principle.   \n\n5) Our results for M86 and M84 suggest that there is no significant interactions taking place between them. Of these four galaxies, only M86 and NGC 4438 appear to be undergoing any noticeable galaxy-galaxy interactions. \n\nDisentangling the interactions of the infalling M86 galaxy group in the directions of M84, NGC 4338, and NGC 4438 will require further observations at multiple wavelengths.  \n\n\n\\section*{Acknowledgments}\nThis work was carried out with proposed and archival observations with {\\it XMM-Newton}, an ESA science mission\nwith instruments and contributions directly funded by ESA Member States and NASA. This research has also made use of the NASA\/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support for this work was provided by the National Aeronautics and \nSpace Administration through Einstein Postdoctoral Fellowship\nAward Number PF9-00070 (A.S.) issued by the Chandra X-ray \nObservatory Center, which is operated for and on behalf of the National Aeronautics and Space\nAdministration under contract NAS8-03060. Further support for this work was provided by the \nDepartment of Energy Grant Number DE-AC02-76SF00515. This work is based [in part] on observations made with Herschel, a\nEuropean Space Agency Cornerstone Mission with significant participation by NASA. Support for this work was provided by NASA through award \\#1428053 issued by JPL\/Caltech.\n\n\n\\bibliographystyle{mnras}\n\\input{reference_defs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{adjustbox}{max width=1.0\\linewidth,center}\n    \\includegraphics{figs\/Intro_cropped.pdf}\n    \\end{adjustbox}\n    \\caption{\\small\n    Results of our approach.\n    (a) Masked input images.\n    (b) Initially restored results from pre-trained inpainting networks in the order of GatedConv\\cite{yu2019free}, EdgeConnect\\cite{nazeri2019edgeconnect}, and GMCNN\\cite{wang2018image}.\n    (c) Results of fine-tuned versions of GatedConv\\cite{yu2019free}, EdgeConnect\\cite{nazeri2019edgeconnect}, and GMCNN\\cite{wang2018image} using the proposed learning algorithm. Restored regions are more natural and consistent with other regions than the initial results.}\n    \\label{fig:intro}\n\\end{figure}\n\n\\label{sec:introduction}\nImage inpainting, a post image processing technique that removes unnecessary parts of images, such as subtitles and obstacles, and fills these areas with new colors and textures, has extensive applications in areas of photo editing, restoration, and even video editing~\\cite{barnes2009patchmatch, levin2004seamless, yu2019free, newson2014video, kim2019deep}.\nThe main goal of inpainting is to generate plausible images that are semantically consistent with the overall context and to fill the missing area to be continuous with the surrounding area.\n\nMany traditional inpainting approaches based on the diffusion technique compute pixel values of the missing area using pixel values in the surrounding area~\\cite{bertalmio2000image, ballester2001filling, levin2003learning}.\nAlthough diffusion-based approaches are effective in recovering small areas (e.g., subtitle removal), they produce blurry results when the missing area is large (e.g., object removal).\nAs an alternative, patch-match-based methods have been proposed to locate similar patches within the given image and copy intensity values of these similar patches into the missing area~\\cite{liang2001real, hays2007scene, barnes2009patchmatch}.\nThis method can fill the missing area with realistic image contents using information available within the input image.\nHowever, it is difficult to restore faces or complex landscapes that require information outside the given image, and it creates irrelevant structures due to the lack of image context inference capabilities.\n\nThe development of deep learning technologies has remarkably improved the performance of image inpainting techniques and made it possible for neural approaches to reconstruct considerably challenging images.\nFirst, context encoder pioneered the use of deep neural networks in the image inpainting task and introduced an architecture composed of convolutional encoder and decoder with adversarial loss~\\cite{pathak2016context, goodfellow2014generative}.\nThe encoder computes a latent feature representation from the input image and the decoder uses the feature representation to restore pixels in the missing area, consequently inferring and reconstructing the overall structure of the input image.\nThe output demonstrates a realistic texture as it approaches the distribution of the real image through the adversarial loss.\nFollow-up studies carry out various attempts while maintaining encoder-decoder structures with discriminators, presented in the work of context encoders~\\cite{pathak2016context}.\nAmong them, attention modules are exploited to generate realistic textures~\\cite{yu2018generative, ren2019structureflow, zeng2019learning, li2020recurrent}.\nThese attention modules allow searching non-local regions to utilize information among similar structures in the given input image and fill pixel values of missing areas.\nSome studies perform structural restoration before carrying out the inpainting to clarify the boundary of the missing hole~\\cite{ren2019structureflow, nazeri2019edgeconnect, xiong2019foreground}.\n\nAlthough these neural approaches trained in a supervised manner with a large external training dataset for the inpainting task have shown promising and satisfying results, they have difficulties in fully utilizing the information within given test images at the test phase.\nThus, we propose a new learning approach to utilize internal statistics available within the given input test image such as patch-recurrence to overcome this limitation.\nPatch-recurrence is a property that many similar patches are existing within a single natural image and numerous image restoration techniques utilize these recurring patches to improve the restoration quality~\\cite{michaeli2014blind, glasner2009super, huang2015single}.\nIn particular, there are several attempts to train the restoration networks for super-resolution and denoising tasks using only internal statistics without the ground-truth clean images~\\cite{shocher2018zero, krull2019noise2void, batson2019noise2self}, and these self-supervised approaches have shown satisfactory results.\nHowever, these self-supervision-based approaches fail to exert the power of deep learning through large external datasets.\n\nWe demonstrate that inpainting networks can be also trained in a self-supervised manner for the first time and develop a new learning algorithm that combines supervised and self-supervised approaches to benefit from both the external dataset and the given input test image in this work.\nThis approach improves the performance of existing state-of-the-art inpainting networks by simply updating network parameters using the internal information (i.e., repeating structure\/texture and color distribution) available from the given input image at test time.\nSpecifically, we use parameters of fully pre-trained inpainting networks on the large external dataset as initial values.\nWe then fine-tune the network parameters in a self-supervised manner at the test stage by exploiting self-similar patches within the input test image and produce improved results as shown in Figure~\\ref{fig:intro}.\nNote that broken lines and crushed edges of the initial result images are clearly restored and the original shapes are properly predicted with the proposed fine-tuning algorithm.\nOur learning method is not restricted to specific network architecture and can be applied to various conventional networks.\nMoreover, our approach can easily upgrade the parameters of the conventional inpainting networks without changing their original architecture.\nThe main contributions of this study are summarized as follows:\n\\begin{itemize}\n\t\\item A novel self-supervised fine-tuning algorithm that automatically takes advantage of recurring patches within the test image is proposed.\n\t\\item Superior inpainting results on benchmark datasets are achieved by utilizing both internal and external datasets.\n\t\\item Our approach can be applied to various inpainting networks without modifying their original network architecture and loss functions.\n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsection{Image inpainting}\n\\begin{comment}\nTraditional image inpainting methods fill the missing area by progressively propagating information from the surrounding area or selecting highly similar patches within the given input image~\\cite{bertalmio2000image,ballester2001filling,levin2003learning,telea2004image,Darabi12:ImageMelding12, hays2007scene}.\nIn particular, patch-based methods~\\cite{Darabi12:ImageMelding12, hays2007scene,  huang2014image} can take advantage of not only local information but also non-local information and can fill the relatively large hole with meaningful content.\nHowever, these methods struggle to create unique content, such as human faces or complex scenes, because they need to meet the assumption that several patches similar to the missing hole in the image are available.\n\\end{comment}\nRecently, deep learning methods have successfully been used in many computer vision tasks, and several learning-based approaches have been proposed for inpainting.\nContext encoder~\\cite{pathak2016context} introduces an encoder\u2013decoder network based on convolutional neural network (CNN) to extract semantics of the image and employs GAN~\\cite{goodfellow2014generative} architecture to generate realistic details.\nIt overcomes the limitation of traditional methods that struggle to create unique content, such as human faces or complex scenes because they fill the missing area using only the information within the given input~\\cite{bertalmio2000image,ballester2001filling,telea2004image,Darabi12:ImageMelding12,barnes2009patchmatch,huang2014image}.\nIizuka et al.~\\cite{iizuka2017globally} removed the pooling layer and employed dilated convolutions~\\cite{yu2015dilated} in the neural networks to solve the problem of the context encoder that generates blurry outputs because of the reduction of the resolution during the encoding stage.\nAlso, they used two discriminators to maintain continuity in and out of the hole.\nThe local one focus on the missing area and the global one assess the entire image.\nYu et al.~\\cite{yu2018generative} analyzed that CNN creates visual artifacts because of the inability to exploit image contents far from the hole.\nThus, contextual attention module is proposed to overcome this limitation.\nContextual attention module scores the similarity of background patches with pixels of the missing area which are estimated in advance by a coarse network.\nThen, patches with high similarity scores are used to reconstruct the missing area.\nGenerative multi-column CNN~\\cite{wang2018image} uses parallel networks with different receptive field sizes to prevent the error propagation caused by the coarse and fine networks that are connected in series.\nIt also uses the reconstruction loss that gives different weights to the pixels in the hole considering the distance from the boundary to solve the spatial-variant constraints.\nLiu et al.~\\cite{liu2018image} proposed the use of partial convolution to learn various forms of masks.\nPrevious methods could not be utilized in practical cases where images were damaged in various forms because they were typically learned with squared masks.\nPartial convolution updates the mask during the inference and renormalizes weights of the masked area every layer to propagate only valid information of the input.\nIn a similar approach by Yu et al.~\\cite{yu2019free}, gated convolution automatically finds a proper mask for the input and considers the gating value calculated across a spatial location of every layer and channel.\nEdgeConnect~\\cite{nazeri2019edgeconnect} introduces a two-stage inpainting network model that first generates the edge map with canny edge detector~\\cite{canny1986computational} and then recovers the whole image.\nHowever, this model is limited by its inability to utilize additional useful information, such as image color.\nStructureFlow~\\cite{ren2019structureflow} uses a smooth image while preserving the edges obtained via RTV~\\cite{xu2012structure} instead of the edge map and employs appearance flow module~\\cite{zhou2016view} to achieve an attention effect for realistic texture.\nLi et al.~\\cite{li2020recurrent} presented recurrent feature reasoning (RFR) network and knowledge consistent attention (KCA) module to cover large holes.\nThey reduce the region of the hole progressively using partial convolution and subsequently merge the inpainting results for each hole.\nAt each iteration, KCA module recurrently calculates the attention score of the missing hole based on the attention score computed in the previous step.\n\n\\subsection{Self-supervised learning for restoration}\nThe self-supervised approach for image restoration primarily refers to the training of the neural networks for the restoration of corrupted images using only information from the input images themselves without corresponding clean images~\\cite{lehtinen2018noise2noise, krull2019noise2void, batson2019noise2self, shocher2018zero}.\nFor the super-resolution task, zero-shot SR (ZSSR)~\\cite{shocher2018zero} trains a small and image-specific CNN during test time and utilizes internal statistics such as patch-recurrence within a given input image.\nHowever, these methods can only utilize internal data because they train the network from random scratch at the test time.\n\nWe present that the self-supervised approach can be applied to image inpainting to fine-tune parameters of networks during test time by exploiting internal statics of a given masked input image.\nMoreover, our method exploits the advantage of both external and internal data by fine-tuning the fully pre-trained networks like~\\cite{park2020fast,lee2020restore} to overcome the limitation of the self-supervised approach that fails to exert the power of deep learning via large external datasets.\n\n\\section{Proposed Method}\nIn this section, we introduce a simple, yet effective self-supervised learning approach to adapt the parameters of pre-trained inpainting networks to the given specific input test image.\n\n\\subsection{Patch-recurrence for inpainting}\nThe internal information available within the input image is very important for the image inpainting task and should be considered to generate semantically consistent and realistic results when filling missing areas.\n\nImages generally demonstrate the property of patch recurrence, which enables the repeated presentation of many identical or similar patches in a single image.\nThe information of similar patches inside the image is used in past patch-match-based methods~\\cite{Darabi12:ImageMelding12, hays2007scene, barnes2009patchmatch, huang2014image} and recent deep-learning methods with the attention mechanism~\\cite{yu2018generative,ren2019structureflow,li2020recurrent}.\nPatch-match-based approaches find similar patches within the target image to fill the hole and paste them.\nBy comparison, the attention module~\\cite{yu2018generative, li2020recurrent} and appearance flow~\\cite{ren2019structureflow} in deep-learning approaches find similar patches\/features in the given input image corresponding to the target (missing) area, and resulting attention maps are used to complete the texture within that area.\nThese methods improve the inpainting results by exploiting the internal information during test time but require an additional network module (e.g., non-local operator) to compute attention maps and are inefficient because they require high computational costs to measure similarity among features at every pixel location.\nSpecifically, $M \\times N$ comparisons are required when the number of pixel values within missing and non-missing areas is $N$ and $M$, respectively, but only a few matches of them produce meaningful high-similarity scores.\n\nTherefore, we introduce a new approach that can fully utilize recurring patches in the given input image during the test phase without explicit similarity comparisons (i.e., patch-match) and also present a new fine-tuning method that enhances the quality of inpainting results of existing networks in a self-supervised manner.\n\n\\subsection{Patch-match-based inpainting without patch-match}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figs\/fig2_alignment_cropped.pdf}\n    \\caption{\\small Inpainting results by EdgeConnect~\\cite{nazeri2019edgeconnect} before and after fine-tuning.\n    (a) Fully pre-trained EdgeConnect~\\cite{nazeri2019edgeconnect} produces artifacts.\n    (b) Fine-tuned EdgeConnect~\\cite{nazeri2019edgeconnect} using a corresponding patch (red box) within the input can remove artifacts.\n    (c) Fine-tuned EdgeConnect~\\cite{nazeri2019edgeconnect} using randomly masked input can also remove artifacts without finding correspondences.\n    }\n    \\label{fig:patchrecurrence}\n\\end{figure*}\n\nNetworks in existing learning-based image inpainting methods learn how to reconstruct the missing area using a large number of training images during the training phase.\nHowever, the input image given at the test phase may include unique and unseen information, such as unusual structures, textures, and colors, which are rare in the training dataset although a large external dataset is available for training.\nIn this case, the pre-trained network may generate an unnatural output incompatible with the specific pattern in the background.\nTherefore, we propose a new inpainting approach to exploit the specific information available within the input test image by utilizing multiple similar patches in the input and further improve the pre-trained network by adapting network parameters to the specific input image.\n\nFigure \\ref{fig:patchrecurrence} illustrates how self-similar patches can be used to fine-tune the network parameters and improve the performance of conventional inpainting networks.\nIn Figure~\\ref{fig:patchrecurrence}(a), we first obtain an initial inpainting result $I_{pred}$ by applying the fully pre-trained EdgeConnect~\\cite{nazeri2019edgeconnect} network $f_{\\theta}$ to an input image $\\tilde{I}$ distorted with a squared mask.\nNext, in Figure~\\ref{fig:patchrecurrence}(b), we then generate a newly masked image $\\tilde{I}_{pred}$ to utilize repeating structures of the given input image for inpainting by removing a self-similar patch in $I_{pred}$ (red squared region).\nThen, by minimizing the MSE between $I_{pred}$ and restored image $f_{\\theta}(\\tilde{I}_{pred})$ from the newly masked image $\\tilde{I}_{pred}$, we can fine-tune the network parameters $\\theta$ to $\\theta^*$.\nAccordingly, we can update network parameters to the input image and can improve the performance of the inpainting network.\nNote that we adapt network parameters without using the ground-truth image during the fine-tuning.\nIn Figure \\ref{fig:patchrecurrence}(c), we show that we can also fine-tune the network by corrupting the initially restored image with random masks without explicit patch-match to find self-similar patches.\nSpecifically, we generate multiple training images for fine-tuning by corrupting $I_{pred}$ using random masks, then fine-tune the network by minimizing MSE between the initial inpainting result $I_{pred}$ and the restored image $f_{\\theta}(A_i)$ from the newly corrupted image $A_i$.\nBy doing so, we demonstrate that we can adapt network parameters without explicitly searching for self-similar patches within the input as in Figure \\ref{fig:patchrecurrence}(b).\n\nTraditional methods require the use of dedicated algorithms and specialized network modules to search similar patches explicitly~\\cite{barnes2009patchmatch, Darabi12:ImageMelding12, hays2007scene, huang2014image}.\nIn contrast, we utilize the self-similar patches by using the randomized masking (corruption) scheme during the test phase without changing the original network architecture.\nMoreover, ours does not require any additional algorithm and\/or network modules to find similar patches (e.g., patch-match).\nAs the area of similar patches can be randomly masked a few times during the randomized masking scheme, these self-similar patches can be naturally exposed to the network during the fine-tuning stage multiple times.\nThus, the inpainting network can learn to restore the missing area using the information from the similar patches, as shown by the result of Figure~\\ref{fig:patchrecurrence}(c), which shows similar improvements to Figure~\\ref{fig:patchrecurrence}(b).\nAnother advantage of this approach is that the randomized-masking-based fine-tuning mechanism does not restrict the shape, size, and number of the recurring patches.\n\n\\begin{algorithm}\n\t\\textbf{Input:} input masked image $\\tilde{I}$, original mask $M$\n\t\n\t\\textbf{Require:} inpainting network $f$ and the pre-trained parameter $\\theta_0$,   number of training $T$, random masks $\\{M_{i}\\}$,\n\tlearning rate $\\alpha$\n\t\n\t\\textbf{Output:} enhanced inpainting result $f_{\\theta^*}(\\tilde{I}, M)$\n\t\n\t\n\t\\nl $i$ $\\leftarrow$ 0\n\t\n\t\\nl $\\theta \\leftarrow \\theta_0$\n\t\n\t\\nl $I_{pred} \\leftarrow f_{\\theta_0}(\\tilde{I}, M)$\n\t\n\t\\While{i $<$ T}{\n\t    \n\t    \\nl $\\tilde{I}_{pred} \\leftarrow I_{pred}\\odot \\left ( 1 - M_i \\right )$\n\t    \n\t    \\nl $I_{pred\\left (\\theta \\right )} \\leftarrow f_{\\theta}(\\tilde{I}_{pred}, M_{i})$\n\t   \n\t    \n\t    \\texttt{\\\\}\n\t    \n\n\t\n\t\n\t\n\t\t\\tcp{\\small Random transformations can be applied for the loss.}\n\t\t\\nl $loss_{rec}(\\theta) \\leftarrow \\|I_{pred}\\odot(1-M)-I_{pred(\\theta)}\\odot(1-M)\\|^2$\n\t\t\n\t\t\\texttt{\\\\}\n\t\t\n\t\n\t\t\\nl $loss_{adv}(\\theta) \\leftarrow \\text{\\small VGG and\/or adversarial~losses}$\n\t\t\n\t\n\t\t\\nl $Loss(\\theta) \\leftarrow loss_{rec}(\\theta) + loss_{adv}(\\theta)$\n\t\t\n        \\texttt{\\\\}\n\t\t\n\t\t\\tcp{\\small Parameter update}\n\t\t\\nl $\\theta \\leftarrow \\theta - \\alpha \\nabla_{\\theta}  \\textit{Loss}(\\theta)$\n\t\t\n\t\t\n       \n\t\t\\nl $i$ $\\leftarrow $ $i$ + 1  \n\t}\n\n\t\\nl $\\theta^* \\leftarrow \\theta$\n\n\t\\textbf{Return:}  $f_{\\theta^*}(\\tilde{I}, M)$\n\t\\caption{\\small \\textit{Fine-tuning algorithm}}\n\t\\label{algorithm}\n\\end{algorithm}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{figs\/overall_flow_cropped.pdf}\n    \\caption{\\small Overall flow of the proposed fine-tuning mechanism.\n    (a) Initial inpainting result from the masked input image using the pre-trained network.\n    (b) Fine-tuning with random masks on the result from the baseline network.\n    (c) Fine-tuned network produces improved result consistent with the input image.}\n    \\label{fig:process}\n\\end{figure*}\n\\subsection{Overall flow}\\label{overall_flow}\n\nThe overall flow of the proposed fine-tuning algorithm is described in Algorithm~\\ref{algorithm} and illustrated in Figure~\\ref{fig:process}.\n\nFirst, we start the learning (fine-tuning) process with the initially restored image $I_{pred}$ similar to~\\cite{lee2020restore} by using the fully pre-trained inpainting network as follows:\n\\begin{equation}\n    I_{pred} = f_{\\theta_0} (\\tilde{I}, M ),\n    \\label{eq3}\n\\end{equation}\nwhere $\\theta_0$ denotes the fully pre-trained parameters of the baseline inpainting network $f$ and $\\tilde{I}$ denotes the masked input image.\nA binary map $M$ denotes a given input mask that corresponds to the input image, where missing and other areas are represented by 1 and 0, respectively.\n\nSecond, we acquire a training dataset using the initially restored image $I_{pred}$.\nWe generate a new and randomly corrupted masked image at the $i_{th}$ fine-tuning iteration as follows:\n\\begin{equation}\n    \\tilde{I}_{pred} = I_{pred}\\odot \\left ( 1 - M_i \\right ),\n    \\label{eq5}\n\\end{equation}\nwhere $M_i$ denotes the randomly generated binary mask and $\\odot$ represents the element-wise multiplication.\nThe newly synthesized masked image $\\tilde{I}_{pred}$ and initially restored image $I_{pred}$ become the input and target of our training dataset, respectively.\nWe render a restored image $I_{pred\\left (\\theta \\right )}$ from the newly masked image $\\tilde{I}_{pred}$ using the inpainting network $f_{\\theta}$.\n\nThird, we compute gradient values with respect to network parameters using the predefined loss functions for pre-training the baseline inpainting network and then update network parameters using a conventional optimizer (e.g., ADAM).\nThe loss is calculated using the difference between $I_{pred}$ and $I_{pred\\left (\\theta \\right )}$.\nNotably, if the baseline inpainting network includes GAN architecture, the discriminator of GAN architecture computes the adversarial loss by using the initially restored image $I_{pred}$ as the real sample rather than the ground-truth clean image.\n\nSpecifically, for the pixel-wise loss (i.e., reconstruction loss), we can use the L2 loss function as follows:\n\\begin{equation}\n    loss_{rec}(\\theta) = \\|I_{pred}\\odot(1-M)- I_{pred(\\theta)}\\odot(1-M)\\|^2.\n    \\label{ExcludeMask}\n\\end{equation}\nIn practice, to compute the reconstruction loss, we exclude the part corresponding to the original mask $M$ (i.e., initially restored area) since we can improve the performance by making this slight modification. Note that, we can use any conventional reconstruction losses beyond the L2 loss (e.g., L1).\nMoreover, when considering additional (perceptual) losses, such as VGG and adversarial losses, we do not need to make any modification. We employ the original function used to train the baseline inpainting networks, \nand the expression is $loss_{adv}(\\theta)$.\nThus, overall loss function that updates parameters of the given inpainting network is expressed as follows:\n\\begin{equation}\n    Loss\\left ( \\theta \\right ) = loss_{rec}(\\theta) + loss_{adv}(\\theta).\n    \\label{OverallLoss}\n\\end{equation}\n\nThen, we repeat these steps $T$ times, which is the number of iterations determined experimentally to achieve best results. Notably, random transformations can be applied during fine-tuning to prevent the network from predicting $I_{pred}$ regardless of its input after adaptation.\n\nFinally, the network parameter is upgraded to $\\theta^*$ and we obtain the final fine-tuned image ${f}_{\\theta^*}(\\tilde{I}, M)$ with the original input image $\\tilde{I}$ and mask $M$.\n\nWe call this algorithm ``restore-from-restored\" because this method leverages the already restored image to improve the inpainting performance.\n\n\\section{Experimental result}\nPlease refer to our supplementary material for more results. \nMoreover, the code, dataset, and pre-trained models for the experiments will be included in the supplementary material upon acceptance.\n\\subsection{Implementation details}\nTo evaluate the performance of the proposed fine-tuning algorithm, we use three different inpainting networks, namely, GatedConv~\\cite{yu2019free}, EdgeConnect~\\cite{nazeri2019edgeconnect}, and GMCNN~\\cite{wang2018image}, as baseline networks of our algorithm.\nThe experiment is performed using the official code for each model.\nNotably, EdgeConnect is currently a state-of-the-art inpainting network.\n\nFor our experiments, we use officially available and fully pre-trained network parameters on the Places2 dataset~\\cite{zhou2017places} for each network and fine-tune the pre-trained parameters via the proposed approach in Algorithm.~\\ref{algorithm}.\nWe evaluate the performance of the proposed algorithm on test sets in conventional benchmark datasets, such as Places2 ~\\cite{zhou2017places} and Urban100~\\cite{huang2015single}.\nThe Places2 dataset is a mixed dataset of people, landscapes, and buildings; the Urban100 dataset mainly consists of buildings with repetitive patterns.\n\nOur random free-form masks used in the fine-tuning process are obtained by the algorithm for sampling free-form masks introduced in \\cite{yu2019free}.\nThe input image size for the fine-tuning is 256$\\times$256 and each input image includes holes covering approximately 20\\% to 40\\% of the image.\nWe utilize the same optimizer and loss function used in each of the pre-trained models, but we modify the learning rate for the fine-tuning: we use learning rate of $10^{-5}$ for EdgeConnect and GatedConv, and $10^{-4}$ for GMCNN.\n\nOur experiments are conducted with Intel i9 and NVIDIA RTX2080 Ti GPU, and it takes 0.2, 0.3, and 0.7 seconds to perform 1 fine-tuning iteration (patch-size = 256$\\times$256, batch-size = 4) with GMCNN, GatedConv, and EdgeConnect respectively.\n\n\n\\subsection{Ablation study}\\label{sec_ablation}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/fig4_ablation_cropped.pdf}\n    \\caption{\\small Visual results on the Urban100 dataset according to the fine-tuning progress.\n    Green, red, and yellow boxes represent the results by GatedConv, EdgeConnect, and GMCNN, respectively.\n    (a) Input masked images.\n    (b) Initially restored image and fine-tuned images for $T$ iterations. \n    \\textbf{Green box}: GatedConv results (From left to right: $T$=0, $T$=200, $T$=400, $T$=700). \\textbf{Red box}: EdgeConnect results (From left to right: $T$=0, $T$=100, $T$=500, $T$=1000).\n    \\textbf{Yellow box}: GMCNN results (From left to right: $T$=0, $T$=50, $T$=150, $T$=250).\n    (c) Ground-truth images.}\n    \\label{fig:iterprogress}\n\\end{figure}\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/urbanG_final_cropped.pdf}\n    \\caption{\\small Improvement of FID and PSNR values of GatedConv, EdgeConnect, and GMCNN with the fine-tuning process on the Urban100 dataset.\n    High PSNR values and low FID scores indicate better quality of images.}\n    \\label{fig:urbangraph}\n\\end{figure}\n\n\n\\paragraph{Number of fine-tuning iterations and image quality.}\nWe show changes in the image quality depending on the number of fine-tuning iterations through the experiment conducted with the Urban100 dataset.\n\nFigure \\ref{fig:iterprogress} illustrates the results of the fine-tuned network with different numbers of iterations for each pre-trained model.\nThe structure of the results is gradually improved as the fine-tuning progress, and the results become close to the ground-truth when it comes to the pre-fixed iteration.\n\nMoreover, Figure \\ref{fig:urbangraph} shows the trend of Fr\u00e9chet inception distance (FID) scores and peak SNR (PSNR) values with a fine-tuning process for each model.\nWe observe that a large number of iterations for the fine-tuning can lead to over-smoothed results depending on the model, and we demonstrate that the FID score can be an important criterion for determining the proper number of iterations $T$ because of its higher sensitivity to over-smoothing than the PSNR value.\n\nIn the experiments of the remaining sections, we use the fixed number of iterations chosen in this manner for fine-tuning.\nSpecifically, the number of fine-tuning iterations is determined after a small random sample experiment performed on each dataset and model.\nNotably, these datasets used to determine $T$ are excluded in our evaluation.\nAs a result, we fine-tune 500, 1000, and 200 iterations on the Places2 test dataset and 700, 1000, and 250 iterations on the Urban100 dataset for each pre-trained model (i.e., GatedConv, EdgeConnect, and GMCNN) at the test phase.\n\n\n\\begin{comment}\n\\subsubsection{Exclude or include the original mask in the loss calculation}\n\\begin{figure}[h]\n    \\centering\n    \\begin{minipage}[b]{55pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/include\/Places365_test_00000045_include_final.png}\\vspace{1pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/include\/Places365_test_00000296_include_final.png}\n    \\end{minipage}\n    \\begin{minipage}[b]{55pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/exclude\/Places365_test_00000045_exclude_final.png}\\vspace{1pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/exclude\/Places365_test_00000296_exclude_final.png}\n    \\end{minipage}\n    \\begin{minipage}[b]{55pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/include\/Places365_test_00000374_include_final.png}\\vspace{1pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/include\/Places365_test_00000598_include_final.png}\n    \\end{minipage}\n    \\begin{minipage}[b]{55pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/exclude\/Places365_test_00000374_exclude_final.png}\\vspace{1pt}\n        \\includegraphics[width=1\\textwidth]{ACCESS\/our_figs\/ablation_study\/qualitative\/exclude\/Places365_test_00000598_exclude_final.png}\n    \\end{minipage}\n    \\caption{\\small Result of fine-tuning with or without the original mask when calculating the loss using the EdgeConnect model.\n    The left side of each image pair shows the case of including the mask, and the right side shows the case of excluding the mask.\n    }\n    \\label{fig:ablationquality}\n\\end{figure}\nAs mentioned in Section \\ref{overall_flow}, we excluded the original mask part when calculating the loss during the fine-tuning step.\nWe perform ablation experiments exploiting the entire baseline image which includes the generated part for the fine-tuning and compare the results with the findings of our method.\nFigure \\ref{fig:ablationquality} presents the comparison of experiments.\nParts of the baseline image generated from the pre-trained network sometimes demonstrate mismatched colors with its surroundings.\nAs a result, colors of generated and surrounding parts are slightly mismatched when these parts are included in the loss calculation at the fine-tuning step, whereas the color of both parts demonstrates satisfactory matching when the mask is excluded.\n\\end{comment}\n\n\n\n\\subsection{Quantitative results}\n\n\\begin{table}[t]\n\\begin{adjustbox}{max width=1.0\\linewidth,center}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline\n\\textbf{Model} & \\textbf{\\# iter.} (=$T$) & \\textbf{PSNR} & \\textbf{SSIM} & \\textbf{L1} (\\%) & \\textbf{FID} & \\textbf{LPIPS} \\\\ \\hline\n\\multirow{5}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Gated\\\\ Conv\\end{tabular}}} & 0 (baseline) & 23.25 & 0.8831 & 2.67 & 60.75 & 0.103 \\\\\n & 100 & 23.10 & 0.8767 & 2.75 & 63.47 & 0.109 \\\\\n & 200 & 23.36 & 0.8820 & 2.69 & 60.70 & 0.103 \\\\\n & 500 & 23.84 & 0.8900 & 2.60 & \\textbf{59.70} & \\textbf{0.097} \\\\\n & 1000 & \\textbf{23.97} & \\textbf{0.8913} & \\textbf{2.60} & 63.51 & 0.101 \\\\ \\hline\n\\multirow{5}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Edge\\\\ Connect\\end{tabular}}} & 0 (baseline) & 24.00 & 0.8934 & 2.84 & 28.06 & 0.098 \\\\\n & 100 & 24.04 & 0.8941 & 2.83 & 27.94 & 0.098 \\\\\n & 200 & 24.06 & 0.8946 & 2.82 & 27.78 & 0.097 \\\\\n & 500 & 24.09 & 0.8952 & 2.81 & 27.53 & 0.097 \\\\\n & 1000 & \\textbf{24.12} & \\textbf{0.8957} & \\textbf{2.80} & \\textbf{27.23} & \\textbf{0.096} \\\\ \\hline\n\\multirow{5}{*}{\\textbf{GMCNN}} & 0 (baseline)& 23.79 & 0.8477 & 6.02 & 69.92 & 0.078 \\\\\n & 100 & \\textbf{24.70} & \\textbf{0.8586} & \\textbf{5.60} & 65.56 & 0.077 \\\\\n & 200 & 24.61 & 0.8550 & 5.74 & \\textbf{61.60} & \\textbf{0.069} \\\\\n & 500 & 24.41 & 0.8475 & 6.05 & 69.57 & 0.073 \\\\\n & 1000 & 24.18 & 0.8439 & 6.31 & 75.13 & 0.082 \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{\\small Fine-tuning results of various inpainting models on the Places2 dataset by changing the number of iterations. PSNR, SSIM, L1, FID, and LPIPS values are measured.}\n\\label{tab:places2-table}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{adjustbox}{max width=1.0\\linewidth,center}\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{} & \\begin{tabular}[c]{@{}c@{}}\\textbf{GatedConv}\\\\ ($T$=700)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\textbf{EdgeConnect}\\\\ ($T$=1000)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\textbf{GMCNN}\\\\ ($T$=250)\\end{tabular} \\\\ \\hline\n\\textbf{PSNR$^\\star$} & 21.38 $\\rightarrow$ 22.90 & 22.73 $\\rightarrow$ 23.19 & 21.64 $\\rightarrow$ 23.42 \\\\ \\hline\n\\textbf{SSIM$^\\star$} & 0.862 $\\rightarrow$ 0.893 & 0.880 $\\rightarrow$ 0.892 & 0.830 $\\rightarrow$ 0.850 \\\\ \\hline\n\\textbf{FID$^\\dagger$} & 75.34 $\\rightarrow$ 58.15 & 53.98 $\\rightarrow$ 45.34 & 77.96 $\\rightarrow$ 70.71 \\\\ \\hline\n\\textbf{LPIPS$^\\dagger$} & 0.096 $\\rightarrow$ 0.078 & 0.085 $\\rightarrow$ 0.073 & 0.083 $\\rightarrow$ 0.082 \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{\\small Changes of PSNR, SSIM, FID, and LPIPS values before and after fine-tuning. GatedConv, EdgeConnect, and GMCNN are fined-tuned on the Urban100 test dataset. The fine-tuning iteration (T) is determined empirically (refer to Sec.~\\ref{sec_ablation}).\nLower $^\\dagger$ and higher $^\\star$ scores indicate better quality of images.}\n\\label{tab:urban-table}\n\\end{table}\n\nWe quantitatively evaluate the performance of fine-tuned networks with L1, peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) to compare the inpainting results objectively (L1 values are provided because it is a common metric in many inpainting studies~\\cite{pathak2016context, nazeri2019edgeconnect, yu2019free}.).\nThese methods mainly measure the distortion of the results, assuming that the ideal results are the same as the original.\n\nMoreover, we compare the perceptual quality of the results by calculating how plausible the produced image looks from the human perspective.\nThis comparison is necessary because the inpainting task does not simply recover images quantitatively but furthermore produces visually appealing images to humans based on the given input image. \nWe use learned perceptual image patch similarity (LPIPS)~\\cite{zhang2018unreasonable} and Fr\u00e9chet inception distance (FID)~\\cite{heusel2017gans} values for the perceptual comparison.\nTo be specific, LPIPS compares the deep features of images extracted from the image classifier based on a CNN architecture, such as VGG~\\cite{simonyan2014vgg} and AlexNet~\\cite{krizhevsky2014alex}.\nWe use AlexNet in this study.\nFID measures the Wasserstein distance (Fr\u00e9chet distance) between distributions of features from the real and generated images.\nThe mean and covariance of each distribution are calculated through the fully pre-trained Inception-V3~\\cite{szegedy2016rethinking} network.\n\nTable~\\ref{tab:places2-table} and Table~\\ref{tab:urban-table} list the quantitative restoration results.\nFirst, Table~\\ref{tab:places2-table} shows the comparison of the results from each fine-tuning iteration on the Places2 dataset.\nOne thousand images from the Places2 test dataset are used for the evaluation.\nWith the aid of our fine-tuning algorithm, overall metrics values improve consistently compared with the restoration results by the pre-trained baseline models (i.e., $T$=0).\nNext, Table \\ref{tab:urban-table} presents the improvements between the results of pre-trained models and the findings after fine-tuning on the Urban100 test dataset.\nAlthough the initial results from the baseline models are poor since the Urban100 dataset is not used in pre-training the baseline networks, our fine-tuning results show considerable improvements.\nThis finding proves that our fine-tuning method enhances the results although the input test image has a slightly different distribution from the dataset used in pre-training.\n\n\n\\subsection{Qualitative results}\n\\begin{figure*}[h]\n    \\centering\n    \\begin{adjustbox}{max width=1.0\\linewidth,center}\n    \\includegraphics{figs\/Qualitative_Results_Urban_cropped.pdf}\n    \\end{adjustbox}\n    \\caption{\\small Qualitative comparison of results on the Urban100 dataset before and after fine-tuning pre-trained models.\n    The first two columns show the ground-truth and masked input image.\n    The next two columns represent image pairs, where the left is the baseline image obtained via the pre-trained inpainting models and the right is the result images by our fine-tuning algorithm.}\n    \\label{fig:qualitative_urban}\n   \n\\end{figure*}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{adjustbox}{max width=1.0\\linewidth,center}\n    \\includegraphics{figs\/Qualitative_Results_Places2_cropped.pdf}\n    \\end{adjustbox}\n    \\caption{\\small Qualitative comparison of results on the Places2 dataset before and after fine-tuning the pre-trained models.\n    A specific part of each result image is magnified to demonstrate improved detail.\n    }\n    \\label{fig:qualitative_places}\n   \n\\end{figure}\n\nWe compare the qualitative results between the initially restored results by the pre-trained models and our results by fine-tuning the pre-trained baselines.\nFigure~\\ref{fig:qualitative_places} shows visual results on the Places2 test dataset.\nNote that, the generated part in the result images of pre-trained models is distorted or does not match the other part.\nBy comparison, our fine-tuned models generate more natural results.\nFurthermore, the visual results are significantly improved if multiple repetitive patterns, such as windows and stairs, exist in the input image since the network is likely to learn the correct answer using many similar patches.\nFigure~\\ref{fig:qualitative_urban} shows the fine-tuning results on the Urban100 dataset.\nThe results on the Urban100 test dataset demonstrate improved performance for structural restoration due to the property of containing many repetitive structures within the image.\n\n\\section{Conclusion}\nA new self-supervision-based inpainting algorithm that allows the adaptation of fully pre-trained network parameters during the test stage is proposed.\nWe utilize self-similar patches within the given input test image to fine-tune the network without using the ground-truth clean image and elevate the performance of networks by combining internal and large external information.\nWe can easily fine-tune the baseline networks and significantly improve the performance over the baselines by optimizing loss functions, which are used to pre-train the baseline networks.\nThe proposed method achieves state-of-the-art inpainting results on the conventional benchmark datasets, and extensive experimental results demonstrate the superiority of our method.\n\n\\if\n\\section*{Acknowledgment}\nThis work was supported by the research fund of Hanyang University (HY-2018).\n\\fi\n\n\\clearpage\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}