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+{"text":"\\section{Introduction}\n\nRapid advances in computer simulations have led to many new developments in \nthe modeling of particulate systems.  \nThese systems represent different real physical systems at different scales, such as  \nthe small  scale of liquid crystals \\cite{Pelzl99}, geological scales of snow and debris \nflow, and the astronomical  scales of planetary rings or dynamics evolution of precursors \nof planets \\cite{poeschel00}. Although particle shape plays an important role, most  \ntheoretical  and numerical developments have been restricted to particles with spherical \nor circular shape. These simplification lead in some cases to unrealistic properties.\nIn collisional non-dissipative systems, spherical (or circular) particles can not \nexchange angular momentum, so that the system cannot explore all the phase space during\nthe evolution.  In dissipative granular systems such as sand piles or fault gouges,\ndisks of spheres tend to roll more easily than non spherical particles, leading to\nunrealistic angles of repose and bulk friction coefficients.\n\nThree different approach has been presented to model the real shape of particulate\nmaterials. In the first approach the shape is represented as a scalar funcions. This\nmodel allows to represent especific shapes, such as ellipses \\cite{ting1993ebd},\nellipsoids \\cite{lin1997tdd} and superquadric \\cite{mustoe2001mfa}.\nThe main drawback of those methods is that the calculations required in the contact\nforce are much more expensive than in spherical (or circular) particles. In the second\napproach the non-spherical particle is represented ad  aggregates or clumps of disks and \nspheres bonded together \\cite{cheng2003des,mcdowell96}. In this approach  crushing and \nfracture of aggregates  can be easily modeled. The disadvantage is that this method requires \na larger number of particles.\n\nThe third approach is represent the complex shape using polygons in 2D,\n\\cite{alonso02a,matuttis00,mirghasemi02}  or polyhedrons in 3D \\cite{mirtich98,cundall1988ftd}\nThe most difficult aspect for the simulations of these objects is the handling of contact interactions. \nNowadays, the interaction is resolved by decomposing them in convex pieces, and applying penalty\n methods, impulse-based methods or dynamic constraints in the interaction between these pieces.  \nImpulse-based methods (also called even-driven methods) allow real-time simulations, but they cannot \nhandle permanent or lasting contacts \\cite{mirtich98}. On the other hand, constraint methods \n(or contact dynamics methods) can handle resting contacts with infinite stiffness, but simulations \nare computationally expensive and lead in some cases to indeterminacy in the solution of contact \nforces . This indeterminacy is removed by using penalty methods, where the bodies are \nallowed to interpenetrate each other and the force is calculated in terms of their overlap. \nHowever, until very recently the determination of such contact force has been heuristic and \nlacks physical correctness, because the interactions do not comply with energy conservation \n\\cite{poeschel04}\n\nWe propose a solution to this problem in 2D based in the mathematical concept of \nspheropolygons.  These objects are generated from the Minkowski  sum of a polygon with a disk, \nwhich is nothing more than the object resulting from sweeping \na disk around the polygon. The 3D counterpart of these objects are the spheropolyhedrons, which\nwere recently introduced by Pournin and Liebling for the simulations of granular media.\nThis simple concept can be used to  generate very complex shapes, including non-convex bodies, \nwithout the need to decompose them  into spherical or convex parts. Here we introduce \nan efficient method to calculate dissipative and non-dissipative interaction between\nspheropolygons. We probe also  that the model complies with the statistical mechanical \nprinciples and the physical laws of a  conservative system. Since the code can simulate \nparticles a wide range of shapes, it can be used  to investigate the effect of the aspect \nratio, angularity and non-convexity of the particles on  granular flow. The relevance of \nthis investigation is demonstrated in the stress-strain response  in biaxial test bellow, \nwhere we show that  the system reach the critical state of soil mechanics, where the particle \nshape strongly affect the material  properties of the granular  media.\n\n\\section{Model}\n\nSystems with different particle shapes are modeled using the concept of the Minkowski\nsum of a polygon (or polyline) with a disk. This mathematical operation is explained\nin Sub-Section \\ref{Minkowski sum}. Sub-section \\ref{mass} leads with the  numerical \ncalculation of mass properties . The interaction between the particles is obtained from \nthe individual interaction  between each vertex of one polygon and each edge of another,\nas explained in Sub-Section \\ref{interaction}. In Sub-Section \\ref{efficient} we present\na method to redocue the number\nof floating point operations used to calculate interaction forces is drastically reduced \nby using a neighbor list and a contact list for each pair of neighbor particles. In\nSub-Section \\ref{time integration} we present the algorith used in each time step of \nthe dynamic simulation.\n\n\n\\subsection{Minkowski sum}\n\\label{Minkowski sum}\nGiven  two sets of points P and Q in an Euclidean space, their Minkowski sum is given  by   \n$P+Q=\\{\\vec x+ \\vec y~|~ \\vec x \\in P,~ \\vec y \\in Q \\}$. \nThis operation is geometrically equivalent to the sweeping of one set around \nthe profile of the other without changing the relative orientation.  A special \ncase is the sum of a polygon with a disk, which is defined here as \nspheropolygon. Other examples of a Minkowski sum are the spherocyllinder \n(sphere + line segment) \\cite{pournin05a}, the spherosimplex  (sphere+simplex) \n\\cite{pournin05b} and the spheropolyhedron (sphere+polyhedron) \\cite{pournin05c},\nwhich are used in simulations of  particulate systems.\n\nThe main advantage of the spheropolygons is that they allow us to represent any\nshape in 2D, from rounded to angular particles, and from convex to non-convex\nshapes. As we will see in Sub-Section \\ref{interaction} The Minkowsky sum does\nnot need to be explicitely calculated to determine the particle interaction. \nThe calculation of the mass properties, however, are calculated numerically, but\nthis does not affect much the simulation time because the calculations are done\nonly a the beginning of the simulations.\n\n\\begin{figure}\n  \\begin{center}\n    \\epsfig{file=grain_shape.eps, width=0.45\\linewidth}\n    \\caption{Minkowski sum of a polygon with a disk.}\n   \\label{fig:Minkowski cows}\n  \\end{center}\n\\end{figure}\n\n\n\\subsection{Mass properties}\n\\label{mass}\n\n\nBefore calculating the mass, center of mass and moment of inertica of the spheropolygons \nwe need to introduce some  useful  concepts. \nGiven a spheropolygon $SP = P+S$, the polygon $P$ will be called the polygon-base; and \nthe radius $r$ of the disk $S$ sphero-radius. The distance \n$d(\\vec X, P)$ from a test point $\\vec X$ to the polygon-base is defined as follows: \nIf the point is outside the polygon, it is given by the minimum\ndistance between the point and the edges of the polygon; If the point is \ninside the polygon, we let $d(\\vec X, P) = 0$. Finally, the point $\\vec X$ is \ninside of the spheropolygon when it satisfies $d(\\vec X, P) < r$.\n \nThe point-inside-spheropolygon test combined with a basic Monte Carlo method \nis used to evaluate the integral expressions for mass, center of mass and the \nmoment of inertia. The numerical integration is performed by taking a \nquasi-random set of points $\\vec X_i$ uniformly distributed in a rectangular \nbox containing the object. Then the integral over the area enclosed by the \nspheropolygon of any function $f(\\vec X)$ is calculated as:\n\n\n\\begin{equation}\n\\mathbf{M}  = \\int_{SP}{f(\\vec X) da} \\approx \\frac{A_{box}}{N_p}\\sum^{N_p}_{i=1}{\\chi(\\vec{X_i})f(\\vec{X_i})}.\n\\label{eq:integral}\n\\end{equation}\n\n\\noindent\n$A_{box}$ is the area of the rectangular box, $\\vec X_i$ is a quasi-random \npoint  inside $A_{box}$, and $N_p=1.6 \\times 10^4$ is the number of points.  $\\chi(\\vec X)$  is the \ncharacteristic function, which returns one if $\\vec X$ is inside the \nspheropolygon  and zero otherwise.  \nReplacing $f(\\vec X)$ by $\\sigma$, $\\sigma\\vec X$ $\\sigma||\\vec X||^2$ results in \n$\\mathbf{M} = m, m \\vec r, I+m||\\vec r||^2$ respectively, where $\\sigma$ is the density, and  $m$, $\\vec r$ and $I$ \nare the mass, center  of mass and moment of inertia.\n\n\\subsection{Interaction force}\n\\label{interaction}\n\nTo solve the interaction between spheropolygons we consider all vertex-edge\ndistances between the polygons base. we consider two spheropolygons $SP_i$ and \n$SP_j$ with their respective polygons base $P_i$ and $P_j$ and sphero-radii \n$r_i$ and $r_j$. Each polygon is defined by the set of vertices $\\{V_i\\}$ and \nedges $\\{E_j\\}$.  The overlapping length between each pair of vertex-edge\n$(V_i,E_j)$ is defined as\n\n\\begin{equation}\n\\delta(V,E)=\\langle r_i+r_j-d(V,E) \\rangle,\n\\label{eq:overlap}\n\\end{equation}\n\n\\noindent\nwhere $d(X,E)= ||\\vec Y-\\vec X||$\nis the Euclidean distance from the vertex $V$ to the segment $E$. Here \n$\\vec X$ is the position of the vertex $V$ and $\\vec Y$ is its closest point \non the edge $E$. The ramp function  $\\langle x\\rangle$ returns $x$ if $x>0$ \nand zero otherwise. The overlapping length in Eq.~(\\ref{eq:overlap})\nis equivalent to the interpenetration between the disks of radii \n$r_i$ and $r_j$ centered on $\\vec X$ and $\\vec Y$.\n\nThe force $\\vec F_{ij}$ acting on particle $i$ \nby the particle $j$ is defined by:\n\n\\begin{equation}\n\\vec F_{ij}=-\\vec F_{ji}=  \\sum_{V_i E_j}{\\vec F(V_i,E_j)}\n               +\\sum_{V_j E_i}{\\vec F(V_j,E_i)},\n\\label{eq:contact force}\n\\end{equation}\n\n\\noindent\nwhere $F(V,E)$ represent the force between the vertex $V$ and the edge $E$. \nif the vertex-edge pair do not overlap, $F(V,E)=0$. Different of\nvertex-edge forces can be included in the modelL linear dashpots,\nnon-linear Hertzian laws, dissipative viscous forces proportional to the\nrelative normal and tangential velocities, sliding friction, etc\nThe force of Eq. \\ref{eq:contact force}  is applied to each particle  in the middle \npoint of the overlap region between the vertex and the edge:\n\n\\begin{equation}\n\\vec R(V,E) = \\vec X + (r_i + \\frac{1}{2} \\delta(V,E)) \n\\frac{\\vec X-\\vec Y}{||\\vec Y-\\vec X||},\n\\end{equation}\n\n\\noindent\nso that the resulting torque on particle $i$ given by $j$ is\n\n\\begin{equation}\n\\begin{array}{clcr}\n\\tau_{ij} & = & \\sum_{V_i E_j}{(\\vec R(V_i,E_j)  - \\vec r_i)\n                        \\times \\vec F(E_i,V_j) } \\\\\n          &+ &\\sum_{V_j E_i}{ (\\vec R(V_j,E_i) - \\vec r_i)\n                        \\times \\vec F(E_j,V_i) }, \n\\end{array}\n\\label{eq:torque}\n\\end{equation}\n\n\\noindent\nwhere $\\vec r_i $  is the center of mass of particle $i$. \n\nThe evolution of  $\\vec r_i$ and the orientation  $\\varphi_i$  of the particle \nis governed by the equations of motion:\n\n\\begin{equation}\n m_i\\ddot{\\vec{r}}_i  =\\sum_{j}\\vec F_{ji}-m_i g \\hat y,  ~~~~~~\nI_i\\ddot{\\varphi}_{i} =\\sum_{j}{\\tau_{ji}}. \n\\label{eq:newton}\n\\end{equation}\n\n\\noindent \nHere $m_i$ and $I_i$ are the mass and moment of inertia of the particle. \nThe sum is over all particles interacting with this particle; $g$ is the\ngravity; and $\\hat y$ is the unit vector along the vertical direction.\n\n\n\\subsection{Efficient calculation of forces} \n\\label{efficient} \n\nThe efficiency of the dynamics simulation is mainly determined  by the method of \ncontact detection. In a system of $N$ particles, each one with $M$ edges, \nthe number of operations required to update the positions of the particle \nin each time step is in the order of $O(N M)$, whereas the number of calculations for \ncontact  detection is $O(N^2 M^2)$. Simulations therefore become very slow when \neither the number of particles or the number of vertices is large.\n\nThe first step to speed up the simulations is to execute the force calculation only  over  \nneighbor particles. With this aim we introduce the  {\\it neighbor list}, which is the\ncollection of pair particles whose distance between them is less than  \n$2\\delta$. \n(The distance between two particles is defined as the minimum of all vertex-edge \ndistances). The parameter $\\delta$ is equivalent to the {\\it Verlet distance}\nused in simulations with spherical particles~\\cite{poeschel04}. As shown in the \nFig.~\\ref{fig:neighbor search}, the Verlet method is equivalent to surround the\nparticles by a {\\it skin}, so that the neighbors list consists of all particles\npairs whose skins overlap. \n\nA {\\it link cell} algorithm \\cite{poeschel04} is \nused to allow rapid calculation of this neighbor  list:   First, the space occupied by  \nthe particles is divided in cells  of side $D+\\delta$, where $D$ is the maximal diameter of \nthe particles. Then  the  link cell list is defined as the list of particles hosted in each cell. \nFinally, the candidates of neighbors for each particle are searched only in the  cell occupied by \nthis particle, and its eight neighbor cells. \n\n\\begin{figure}[t]\n  \\begin{center}\n    \\epsfig{file= linkcell,width=\\linewidth}\n   \\caption{Method for identification of neighbor list: the space domain is divided by square cells.\nThen the potential neighbor of the particles are those hosted in the same cells, or in the adjacent\ncells.  Each particle has as {\\it skin} of thickness $\\delta$. If the skins of two potential neighbors \noverlap, they are included in the neighbor list.\n}\n   \\label{fig:neighbor search}\n  \\end{center}\n\\end{figure}\n\nThe neighbor list is calculated at the beginning of the simulation, and it is \nupdated when the following condition is satisfied:\n\n\\begin{equation}\n \\max_{1 \\le i \\le N}\\{\\Delta x_i+R_i\\Delta\\theta_i\\}>\\delta.\n\\label{eq:neighbor update condition}\n\\end{equation}\n\n\\noindent\n$\\Delta x_i$ and $\\Delta\\theta_i$ are the maximal displacement and rotation\nof the particle after the last neighbor list update. $R_i$ the maximal distance \nfrom the points on the particle to its center of mass.  After each update \n$\\Delta x_i$ and $\\Delta\\theta_i$ are set to zero. The update condition is\nchecked in each time step. Increasing the value of  $\\delta$ makes  updating \nof the list less frequent, but increases its size, and \nhence the memory  used in the simulation. Therefore, the parameter $\\delta$ must \nbe chosen by making a compromise between the storage (size of the neighbor list) and \nthe computer time (frequency of list updates). \n\nNeighbor list reduces the amount of calculations to  $O(N M^2)$.  \nTherefore the simulations are still very expensive when particles \nconsist of a large number of vertices. Further reduction of the number of calculations \nbetween neighbor particles can be done by identifying which part of a particle \nis neighbor to the other. This idea is implemented as follows:  for each element\nof the neighbor list, we create a {\\it contact list}, which consists of those \nvertex-edge pairs whose distance between them is less than $r_i+r_j+2\\delta$,\nwhere $r_i$ and $r_j$ are the sphero-radii. \nIn each time step, only these vertex-edge pairs are involved in the contact \nforce calculations. Overall, neighborhood  identification requires a neighbor list \nwith all pair of neighbor particles, and one contact list for each pair of neighbors. \nThese lists require little memory storage, and they reduce the amount of calculations \nof contact forces to $O(N)$, which is of the same order as in simulations with \nspherical particles \\cite{poeschel04}.\n\n\\subsection{Time Integration}\n\\label{time integration}\nThe equations of motion of the system are numerically solved using a four order \npredictor-corrector  algorithm \\cite{poeschel04}. A pseudocode with the basic procedures \nin each time step is shown in Algorithm  \\ref{alg: one step}.  The predicted method calculates the position \n(center of mass and orientation) of each particle and its derivatives using a Taylor expansion. \nNext the vertices of the all polygons are updated according to the predicted positions of the particles.\nIf the neighbor update condition of Eq.~\\ref{eq:neighbor update condition} is satisfied, the link cell \nis calculated, and then it is used to update the neighbor list and the contact list of each one of its \nelements. Then the contact forces and torques are calculated. Finally, forces and torques are used to \ncorrect the position of the particles and their derivatives. The algorithm is basically the same as the \nused in polygons, except that the force is calculated using Eqs.~\\ref{eq:contact force}.  \nNote that the efficiency of the code is based in the simplicity of the contact force calculation, and \nin the fact that the Minkowski sum does not  need to be calculated during the time integration.\n\nThe parameters of the simulations are a constant stiffness $k=10^7 N \/m $, \ngravity $g=10m\/s^2$, density $\\sigma=1kg\/m^2$, time step $\\Delta t = 10^{-5}s$ \nand Verlet distance $\\delta = 1m$.\n\n\\begin{algorithm}[H]\n\\SetLine\n\\KwIn{state of the particles at time $t$}\n\\KwOut{state of the particles at time $t+\\Delta t$ }\n\npredict position of the particles and its derivatives\\;\n\t\n\\If{neighbor update condition is satisfied}\n    {  calculate link cell\\;\n       update neighbor list\\;\n       update contact lists\\;\n    }\nupdate vertices of the particles\\;\ncalculate contact forces between neighbor particles\\;\napply contact forces to the particles\\;\napply gravity forces to the particles\\;\ncorrect positions and its derivatives using forces and torques\\;\n\\\n\\caption{One time step of the time integration scheme}\n\\label{alg: one step}\n\\end{algorithm}\n\n\\section{Non-dissipative granular dynamics simulations}\nAs a first step we present here simulations results of many body conservative systems.\nWe investigate the evolutions towards the statistical mechanical equilibrium of this\nsimple system. Generalization to dissipative system driven by external forces will\nbe presented in forthcoming papers.\n\n\\subsection{Energy balance}\nThe vertex-edge interaction between the particles is given by\n\n\\begin{equation}\n\\vec F(V,E) = k \\delta(V,E) \\vec N\n\\label{eq:elastic force}\n\\end{equation}\n\n\\noindent\nwhere The material parameter $k$ is the stiffness constant,\n$\\delta(V,E)$ is given by Eq.~\\ref{eq:overlap} and \n$\\vec N$ is the unit normal vector:\n\n\\begin{equation}\n\\vec N = \\frac{\\vec Y-\\vec X}{||\\vec Y-\\vec X||}\n\\label{eq:normal}\n\\end{equation}\n\nHere  $\\vec X$ is the position of the vertex $V$ and $\\vec Y$ is its closest point \non the edge $E$.\n\nThe question which now arises is whether the vertex-edge interaction in\nEq.~\\ref{eq:elastic force}  leads to a  conservative system. Let us multiply the \nfirst equation in  ~(\\ref{eq:newton}) by  $\\dot{\\vec{r}}$ and the second one by \n$\\dot\\varphi$. Next we sum both equations and then sum over all particles. \nAfter some algebra we get the energy conservation equation:\n\n\\begin{equation}\nE_T=\\sum_{ij}{E^e_{ij}}+\\sum_i{(\\frac{1}{2} m_i v^2_i + \\frac{1}{2} I_i \\omega^2_i)} = cte.\n\\label{eq:energy}\n\\end{equation}\n\n\\noindent\nThe first term of this equation corresponds to the potential elastic energy \nat the contacts:\n\n\\begin{equation}\nE^e_{ij} =  \\frac{1}{2} k (\\sum_{V_i E_j}{\\delta(V_i,E_j) } \n        +\\sum_{V_j E_i}{ \\delta(V_j,E_i)\\ }).\n\\label{eq:potential}\n\\end{equation}\n\n\\noindent\nThe other terms of Eq.~(\\ref{eq:energy}) are \nthe linear and rotational kinetic energy of the particles. We emphasized \nthat the elastic force in Eq.~(\\ref{eq:contact force}) belongs to the potential \nenergy  defined by Eq.~(\\ref{eq:potential}), which proves that our model is \nconservative. The  simplicity of this force contrasts to the P\\\"oschel's model \nfor interacting triangles \\cite{poeschel04}, where the forces and torques associated \nto his potential energy lead to a much more expensive calculation. \n\n\\begin{figure}\n  \\begin{center}\n       (a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)\\\\\n    \\epsfig{file=disks_loose_b.eps,width=0.45\\linewidth}~~~\n    \\epsfig{file=rice_loose_b.eps, width=0.45\\linewidth}\n       (c)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(d)\\\\\n    \\epsfig{file=peanuts_loose_b.eps, width=0.45\\linewidth}\n    \\epsfig{file=pebbles_loose_b.eps, width=0.45\\linewidth}\n   \\caption{\n Systems obtained from Minkowski sum approach:\n(a) disks (Vertex + disk); (b) rice (segment + disk);\n(c) peanuts (Polyline + disk) and (d) pebbles (triangle +disk)\n}\n   \\label{fig:Minkowski sum}\n  \\end{center}\n\\end{figure}\n\n\nOther important aspect of this dynamics simulation is the accuracy of the numerical \nsolution.  The numerical error in the energy calculation is evaluated by performing \na series of simulations with many  non-spherical particles interaction via the elastic\nforce given by Eq.~\\ref{eq:elastic force}. Each test consists of $400$ particles \nconfined by four fixed rectangular  walls. Each particle occupies an area of $1cm^2$ and \nthe confining area is $46cm\\times46.25cm$. These dimensions lead to a volume  fraction \nof $\\Phi = 0.186$. Each sample  consists of identical particles  with a specific \nspheropolygonal shape  as shown   Fig.~\\ref{fig:Minkowski sum}: disks (point+disks) \nrice (line+disk),  peanuts (polyline+disk), and pebbles (triangle+disk).\n\nInitially, each particle has zero angular velocity and a linear velocity of \n$1cm\/s$ with  random orientation. Due to collisions,  the linear momentum of each \nparticle changes and part of it is transferred to angular momentum. Fig.\\ref{fig:energy}\nshow the potential (a) kinetic (b) and total (c) energy of the system. Elastic energy \nhas a negligible contribution to the energy budget, as it differs from zero only for short \ntimes  during collisions. Energy conservation is numerically verified within a porcentual \nerror of $0.01\\%$. The energy fluctuations are produced by time discretization\nWe shall note alse that energy have a trend to grow slowly in all samples.\n\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\epsfig{file=total_energy.eps,width=\\linewidth}\n   \\caption{Time evolution of the total kinetic and potential energy in\nthe non-dissipative system.  As is expected from the energy conservation,\nthe total energy keeps constants, except numerical error wicho are lower than\n$0.01\\%$ thoughout all the simulations.\n}\n   \\label{fig:energy}\n  \\end{center}\n\\end{figure}\n\n\\subsection{Statistical equllibrium}\n\nSimulations with a large number of particles show that elastic interactions\nallow the particles to exchange energy and momentum, whereas its contribution to the \nenergy budget is negligible. This property leads us to investigate the existence of \na statistical equilibrium  in a gas of  non-spherical particles.  It is expected that \nthe system will reach the statistical equilibrium, which is characterized by an\nenergy equipartition and a   Maxwell-Boltzmann statistics for energy  distribution \n\\cite{tolman}:\n\nThe energy of the system consist of rotational and translational kinetic energy. At \nthe beginning of the simulations the  rotational kinetic energy is zero, and it \nincreases during the simulations due to collision, see left part of the  F\nig.~\\ref{fig:statistical equilibrium}.   For all the samples, we observe the same \nstationary  regime, where  the average of   rotational kinetic energy reaches the \nlimit of $1\/2$  of the linear kinetic energy.   This is in agreement with the \ntheorem of equipartition  of energy \\cite{tolman},  which states that each \nquadratic term in the energy should  contribute the same  weight in the mean energy \nof the system.\n.\nIt is expected that the system reach the statistical equilibrium, which is\ncharacterized  by a  Maxwell-Boltzmann statistics for energy  distribution \n\\cite{tolman}:\n\n\\begin{equation}\n\\rho(E_k)dE_k = 2\\sqrt{E_k\/\\pi(kT)^3} \\exp(-E_k\/k_BT)dE_k,\n\\label{eq:Maxwell-Boltzmann}\n\\end{equation}\n\n\\noindent\nhere $E_k=\\frac{1}{2}(mv^2_x+mv^2_y+I\\omega^2)$ is the kinetic energy of the particle;\n$T$ the temperature; and $k_B$ the Boltzmann constant. The mean energy of the \nparticles leads to $\\bar E_k=\\int^\\infty_0{\\rho(E_k)E_kdE_k}=\\frac{3}{2}k_B T$.\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\epsfig{file=equipartition_energy.eps,width=\\linewidth}\n   \\caption{Time evolution of the total linear and rotational\nkinetic energy of the particles. $E_0$ is the initial value of the total \nkinetic energy. The horizontal lines correspond to the expected value \nby the equilibrium statistical mechanics.}\n   \\label{fig:equipartition}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\epsfig{file=energy_distribution_d.eps,width=\\linewidth}\n   \\caption{energy distribution $N(\\epsilon)$ for the particles, where \n$\\epsilon=E_k\/\\bar E_k$.The line corresponds to the best fit \n$n(\\epsilon)= 2\\beta \\sqrt{\\epsilon\\beta\/\\pi} \\exp(-\\beta \\epsilon)$,\nwith $\\beta = 1.5$.\n}\n   \\label{fig:energy distribution}\n  \\end{center}\n\\end{figure}\n\nWe also calculate the energy distribution of the particles in the\nstationary regime. In order to calculate the energy distribution, we take \nsnapshots of the simulations between $t=1s$ and $t=8s$ distanced by $0.01s$.  \nIn each snapshot the kinetic energy of the individual particles is measured. The \nhistogram of the  variable $\\epsilon = E_k\/\\bar E_k$ is constructed using $100$ identical bins between \nzero and the maximal value. According to  Eq.~(\\ref{eq:Maxwell-Boltzmann}), the \ntheoretical distribution of $\\epsilon$ must satisfy\n$\\rho(\\epsilon)d\\epsilon = 2\\sqrt{\\epsilon \\beta^3} \\exp(-\\beta \\epsilon)d\\epsilon$,\nwhere $\\beta = 3\/2$. An excellent agreement between this theoretical distribution\nand the simulations data is shown in Fig.~\\ref{fig:energy distribution}. Simulations\nwith different non-spherical shapes, which will be presented elsewhere \\cite{alonso08a}, \nshow that energy distribution for all samples collapse onto the theoretical expected value. \nIt is also shown that the relaxation time for the statistical equilibrium is very sensitive \nto the degree of non-sphericity of the particle.\n\n\\section{Dissipative granular dynamics simulation}\n\nHere we present biaxial tests simultions with circular and non-circular particles.\nUsually, the granular assemblies are compacted and loaded within a \nset of confining walls. These walls act as boundary conditions, \nand can be moved by specifying their velocity or the force applied \non them. The response of the walls can be used to calculate the \nglobal stress and strain of the assembly.\n\nThe interaction of the spheropolygonspolygons with the walls is modeled here by \nusing a simple visco-elastic force. First, we allow the polygons \nto penetrate the walls. Then, for each vertex of the polygon $\\alpha$\ninside of the walls we include a force\n\n\nConfining walls can be used to generate samples with different void\nratios. Starting from a very loose packing, the sample is compacted by \napplying a centripetal gravitational field to the particles and on the \nwalls, oriented to the center of mass of the assembly. Then the sample \nis subjected to an isotropic compression until the desired confining \npressure is reached. In order to generate dense samples, the interparticle \nfriction is set to zero during the construction. The loose samples are created \ntaking damping coefficients 100 times greater than those used in the test \nstage. Samples with void ratio ranged from $0.128$ to $0.271$ \ncan be achieved with this method \\cite{pena07}.\n\nWe have investigated shear deformation of granular samples with \ndifferent initial void ratios~\\cite{pena07}. Shear bands are observed in \ndense samples, whereas they seems to be absent in loose ones. they share \nsome common properties of the shear bands in real granular materials,\nsuch as their characteristic reflection when they reach the boundary wall.  \nShear band orientation lies between the Roscoe angle and Mohr-Coulomb \nsolution, as in most of the experimental data.\n\nFor large shear deformations all samples reaches the critical state, which\nis independent on their initial density. Once the samples reaches this \nstate, they deform  at constant void ratio and coordination number~\\cite{pena07}. The \nevolution of  the deviatoric stress exhibits fluctuations around the residual \nstrength. Abrupt reduction of the stress results from the collapse of \nforce chains, as shown the Fig.\\ref{Fig:Collapse_fch}. collapse of \nforce chains makes the sample to approach and retreats unstable\nstages. A similar  behavior is observed  in glass bead samples \n\\cite{nasuno97} and packings of \nglass spheres \\cite{adjemian05}. Experimental biaxial tests show evidence of \n\\textsl{dynamic instabilities} at the critical state \\cite{vardoulakis05}. \nErratic slip-stick motion at the critical state is interesting, owing to  \nits potential analogy with earthquake dynamics \\cite{alonso06}.\n\n\n\\begin{figure}\n  \\begin{center}\n       (a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)\\\\\n    \\epsfig{file=disks_dense_b.eps,width=0.45\\linewidth}~~~\n    \\epsfig{file=rice_dense_b.eps, width=0.45\\linewidth}\n       (c)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(d)\\\\\n    \\epsfig{file=peanuts_dense_b.eps, width=0.45\\linewidth}\n    \\epsfig{file=pebbles_dense_b.eps, width=0.45\\linewidth}\n   \\caption{\nGranular packing obtained with disks, rice, peanuts and rice\n}\n   \\label{fig:Minkowski cows}\n  \\end{center}\n\\end{figure}\n\n\\section{Performance}\nLastly, We compare the efficiency of the many-body simulations of systems consisting \non disks, spheropolygons  and clums of spheres. Each spheropolygons correspond to a \nparticle with complex shape, and it consist on $62$ vertices.The clump of spheres represents \nthe same complex shaped particles, and it need $726$ particles. The macroparticles \nthey are simulating by summing up the contact forces between the constituting disks, and  updating all the disks of \neach particle according with its current position.\nThe performance of the simulations is estimated by  running different processes in a \nPentium 4, 3.0GHz, and calculating  the  {\\it Cundall number} is each one of them. \nThis number is the amount of particle time  steps executed by the processor in one second, \nwhich is calculated as  $c = N_T N \/T_{CPU}$, where $N_T$ is the number of time steps, $N$ is the \nnumber of particles and $T_{CPU}$ is the CPU time of the simulation.  \nFig.~\\ref{fig:performance}~shows the Cundall number versus the\nnumber of particles for the three  cases. \nWhen the number of particles is between $N = 100$ and $N=1000$ the Cundall number is \napproximately constant. This constant  is around  $100,000$ for disks, \n$2,000$ for spheropolygons, and $50$ for clumpy particles. Therefore the simulations\nwith spheropolygons, although slower than simulations with disks, are much\nfaster than simulations with clumpy particles.  This is because each time step needs to \nupdate the position of  $62$ vertices  of the shperopolygons  whereas it needs to \nupdate the position of the $726$ disks in the  case of the clumpy particles.  Therefore \nsimulations with spheropolygons are more efficient than those ones with clumps of \ndisks, because the former ones require less elements to represent the particle \nshape.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\epsfig{file=performance.eps,width=\\linewidth}\n   \\caption{Cundall number versus the number of particles, in simulations with\ndisks, spheropolygons and clumpy particles.\n}\n   \\label{fig:performance}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n      (a)\\\\\n    \\epsfig{file=stress_strain_shape,width=\\linewidth}\\\\\n    (b)\\\\\n    \\epsfig{file=voidratio_strain_shape,width=\\linewidth}\n   \\caption{Stress versus strain (a) and void ration versus strain (b) for different \n            particles shape, in biaxial test simulations.The dotted line in (a) represents\n            $p\/k_n$, where p is the applied pressure on the latteral walls.}\n   \\label{fig:performance}\n  \\end{center}\n\\end{figure}\n\n\\section{Concluding remarks}\n\nThe  method presented here provide an energy balance equations and a wide \nrange of particle shape representations, \nincluding non-convex particles and tunable grain roundness.\nThis paper shows that modeling interacting particles using spheropolygons has \nseveral advantages  with respect to other existing particle-based models: \ni) The possibility to model non-convex particles; ii) a realistic representation of the surface \ncurvature of particles; iii) guaranteed compliance with physical and \nstatistical mechanical laws; iv) balance between accuracy and efficiency. \nBenchmark tests prove energy conservation with an error below $0.0001\\%$. \nSimulations with many particles verify the Maxwell-Boltzmann distribution and the principle\nof energy equipartition. The computational efficiency is compared to simulations with disks\nand clumps of disks. Simulations with disks are around $50$ times faster than simulation \nwith spheropolygons.  However, the speed of the simulation is $40$ times faster that \nsimulations with clumpy particles.\n\nTHe method overcome also two main difficulties existing in previous DEM developments.\n\n(1) The interaction between polygons using the overlapping area is \ndifficult to generalize in 3D, because  the overlap between polyhedrons \nis much more difficult to evaluate.\n\n(2) The elastic force used in this work does not belong from a \npotential, so that this model does not provide an equation for \nenergy balance. In the investigation of fault zones, the energy \nbalance is required  to determine the energy  budget in earthquakes.\n\nThe model is still very simple, but extensions to more complex interactions\nand 3D simulations are achievable in the near future. Cohesive and frictional forces \ncan be incorporated by introducing internal variables in each vertex-edge contact.\nThese variables account for elastic deformation at the contacts and they must be\nupdated in each time step according to the sliding conditions or breaking criterion. \n3D modeling using spheropolyhedra requires elastic forces similar to \nEq.~(\\ref{eq:contact force}), where the sum is over all vertex-face  and edge-edge \ninteractions. Special attention is required  for the case of two parallel edges in contact. \nThis case lead to a non-uniqueness in the  selection of contact points. This need to be \nresolved to get a physical correctness in the torque calculation.\n   \n\nI thank Syed Imran for technical support;\nR. Cruz-Hidalgo and K. Steube for review of an early version of \nthe manuscript; A.J. Hale and M. L. Kettle for writing corrections;\nand S. Latham, Weatherley, E. Heesen,  H. Muhlhaus,  P. Mora, W. Hancock, \nP. Clearly, S. Luding, and S. McNamara for  discussions. This work is supported by\nthe Australian Research Council (project number DP0772409 ) and the AuScope project.\n\n\\bibliographystyle{elsart-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex}\n {2.3ex plus .2ex}{\\large\\bf}}\n\\def\\@startsection{subsection}{2}{\\z@}{2.3ex plus .2ex{\\@startsection{subsection}{2}{\\z@}{2.3ex plus .2ex}\n {2.3ex plus .2ex}{\\bf}}\n\\newcommand\\Appendix[1]{\\def\\Alph{section}}{Appendix \\Alph{section}}\n \\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{\\label{#1}}\\def\\Alph{section}}{\\Alph{section}}}\n \n\n\n\\newcommand{\\oo}[2]{\\left(#1\\left|#2\\right.\\right)}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\begin{document}\n\n\n\\begin{titlepage}\n\\samepage{\n\\setcounter{page}{1}\n\\rightline{LTH--895}\n\\rightline{April 2011}\n\n\\vfill\n\\begin{center}\n {\\Large \\bf Top Quark Mass in \\\\\nExophobic Pati--Salam Heterotic String Model\n}\n\\vspace{1cm}\n\\vfill {\\large\nKyriakos Christodoulides$^{1}$,\nAlon E. Faraggi$^{1}$,\n and\nJohn Rizos$^{2}$}\\\\\n\\vspace{1cm}\n{\\it $^{1}$ Dept.\\ of Mathematical Sciences,\n             University of Liverpool,\n         Liverpool L69 7ZL, UK\\\\}\n\\vspace{.05in}\n{\\it $^{2}$ Department of Physics,\n              University of Ioannina, GR45110 Ioannina, Greece\\\\}\n\\vspace{.025in}\n\\end{center}\n\\vfill\n\\begin{abstract}\n\nWe analyse the phenomenology of an exemplary exophobic\nPati--Salam heterotic string vacuum, in which\nno exotic fractionally charged states exist in the massless string spectrum.\nOur model also contains the\nHiggs representations that are needed to break the gauge symmetry\nto that of the Standard Model and to generate fermion masses\nat the electroweak scale.\nWe show that the requirement of a leading mass term for the\nheavy generation,  which is not degenerate with the mass\nterms of the lighter generations, places\nan additional strong constraint on the viability of the models.\nIn many models a top quark Yukawa may not exist at all, whereas\nin others two or more generations may obtain a mass term at leading\norder. In our exemplary model  a mass term at leading\norder exist only for one family. Additionally,\nwe demonstrate the existence of supersymmetric $F$-- and $D$--flat\ndirections that give heavy mass to all the\ncolour triplets beyond those of the Standard Model\nand leave one pair of electroweak Higgs doublets light.\nHence, below the Pati--Salam breaking scale, the matter states in our\nmodel that are charged under the observable gauge symmetries,\nconsist solely of those of the Minimal Supersymmetric Standard Model.\n\n\n\\noindent\n\n\\end{abstract}\n\\smallskip}\n\\end{titlepage}\n\n\\setcounter{footnote}{0}\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\n\\def\\noindent {\\noindent }\n\\def\\nonumber {\\nonumber }\n\\def{\\it i.e.}{{\\it i.e.}}\n\\def{\\it e.g.}{{\\it e.g.}}\n\\def{\\textstyle{1\\over 2}}{{\\textstyle{1\\over 2}}}\n\\def{\\textstyle {1\\over3}}{{\\textstyle {1\\over3}}}\n\\def{\\textstyle {1\\over4}}{{\\textstyle {1\\over4}}}\n\\def{\\textstyle {1\\over6}}{{\\textstyle {1\\over6}}}\n\\def{\\tt -}{{\\tt -}}\n\\def{\\tt +}{{\\tt +}}\n\n\\def{\\rm Tr}\\, {{\\rm Tr}\\, }\n\\def{\\rm tr}\\, {{\\rm tr}\\, }\n\n\\def\\slash#1{#1\\hskip-6pt\/\\hskip6pt}\n\\def\\slash{k}{\\slash{k}}\n\\def\\,{\\rm GeV}{\\,{\\rm GeV}}\n\\def\\,{\\rm TeV}{\\,{\\rm TeV}}\n\\def\\,{\\rm y}{\\,{\\rm y}}\n\\defStandard--Model {Standard--Model }\n\\defsupersymmetry {supersymmetry }\n\\defsupersymmetric standard model{supersymmetric standard model}\n\\def\\vev#1{\\left\\langle #1\\right\\rangle}\n\\def\\langle{\\langle}\n\\def\\rangle{\\rangle}\n\\def\\o#1{\\frac{1}{#1}}\n\n\\def{\\tilde H}{{\\tilde H}}\n\\def{\\overline{\\chi}}{{\\overline{\\chi}}}\n\\def{\\overline{q}}{{\\overline{q}}}\n\\def{\\overline{\\imath}}{{\\overline{\\imath}}}\n\\def{\\overline{\\jmath}}{{\\overline{\\jmath}}}\n\\def{\\overline{H}}{{\\overline{H}}}\n\\def{\\overline{Q}}{{\\overline{Q}}}\n\\def{\\overline{a}}{{\\overline{a}}}\n\\def{\\overline{\\alpha}}{{\\overline{\\alpha}}}\n\\def{\\overline{\\beta}}{{\\overline{\\beta}}}\n\\def{ \\tau_2 }{{ \\tau_2 }}\n\\def{ \\vartheta_2 }{{ \\vartheta_2 }}\n\\def{ \\vartheta_3 }{{ \\vartheta_3 }}\n\\def{ \\vartheta_4 }{{ \\vartheta_4 }}\n\\def{\\vartheta_2}{{\\vartheta_2}}\n\\def{\\vartheta_3}{{\\vartheta_3}}\n\\def{\\vartheta_4}{{\\vartheta_4}}\n\\def{\\vartheta_i}{{\\vartheta_i}}\n\\def{\\vartheta_j}{{\\vartheta_j}}\n\\def{\\vartheta_k}{{\\vartheta_k}}\n\\def{\\cal F}{{\\cal F}}\n\\def\\smallmatrix#1#2#3#4{{ {{#1}~{#2}\\choose{#3}~{#4}} }}\n\\def{\\alpha\\beta}{{\\alpha\\beta}}\n\\def{ (M^{-1}_\\ab)_{ij} }{{ (M^{-1}_{\\alpha\\beta})_{ij} }}\n\\def{\\bf 1}{{\\bf 1}}\n\\def{(i)}{{(i)}}\n\\def{\\bf V}{{\\bf V}}\n\\def{\\bf N}{{\\bf N}}\n\n\\def{\\bf b}{{\\bf b}}\n\\def{\\bf S}{{\\bf S}}\n\\def{\\bf X}{{\\bf X}}\n\\def{\\bf I}{{\\bf I}}\n\\def{\\mathbf b}{{\\mathbf b}}\n\\def{\\mathbf S}{{\\mathbf S}}\n\\def{\\mathbf X}{{\\mathbf X}}\n\\def{\\mathbf I}{{\\mathbf I}}\n\\def{\\mathbf \\alpha}{{\\mathbf \\alpha}}\n\\def{\\mathbf \\beta}{{\\mathbf \\beta}}\n\\def{\\mathbf \\gamma}{{\\mathbf \\gamma}}\n\\def{\\mathbf \\xi}{{\\mathbf \\xi}}\n\n\\def\\t#1#2{{ \\Theta\\left\\lbrack \\matrix{ {#1}\\cr {#2}\\cr }\\right\\rbrack }}\n\\def\\C#1#2{{ C\\left\\lbrack \\matrix{ {#1}\\cr {#2}\\cr }\\right\\rbrack }}\n\\def\\tp#1#2{{ \\Theta'\\left\\lbrack \\matrix{ {#1}\\cr {#2}\\cr }\\right\\rbrack }}\n\\def\\tpp#1#2{{ \\Theta''\\left\\lbrack \\matrix{ {#1}\\cr {#2}\\cr }\\right\\rbrack }}\n\\def\\langle{\\langle}\n\\def\\rangle{\\rangle}\n\\newcommand{\\cc}[2]{c{#1\\atopwithdelims[]#2}}\n\\newcommand{\\nonumber}{\\nonumber}\n\n\n\\def\\,\\vrule height1.5ex width.4pt depth0pt{\\,\\vrule height1.5ex width.4pt depth0pt}\n\n\\def\\relax\\hbox{$\\inbar\\kern-.3em{\\rm C}$}{\\relax\\hbox{$\\,\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm C}$}}\n\\def\\relax\\hbox{$\\inbar\\kern-.3em{\\rm Q}$}{\\relax\\hbox{$\\,\\vrule height1.5ex width.4pt depth0pt\\kern-.3em{\\rm Q}$}}\n\\def\\relax{\\rm I\\kern-.18em R}{\\relax{\\rm I\\kern-.18em R}}\n \\font\\cmss=cmss10 \\font\\cmsss=cmss10 at 7pt\n\\def\\IZ{\\relax\\ifmmode\\mathchoice\n {\\hbox{\\cmss Z\\kern-.4em Z}}{\\hbox{\\cmss Z\\kern-.4em Z}}\n {\\lower.9pt\\hbox{\\cmsss Z\\kern-.4em Z}}\n {\\lower1.2pt\\hbox{\\cmsss Z\\kern-.4em Z}}\\else{\\cmss Z\\kern-.4em Z}\\fi}\n\n\\defA.E. Faraggi{A.E. Faraggi}\n\\def\\JHEP#1#2#3{{\\it JHEP}\\\/ {\\bf #1} (#2) #3}\n\\def\\NPB#1#2#3{{\\it Nucl.\\ Phys.}\\\/ {\\bf B#1} (#2) #3}\n\\def\\PLB#1#2#3{{\\it Phys.\\ Lett.}\\\/ {\\bf B#1} (#2) #3}\n\\def\\PRD#1#2#3{{\\it Phys.\\ Rev.}\\\/ {\\bf D#1} (#2) #3}\n\\def\\PRL#1#2#3{{\\it Phys.\\ Rev.\\ Lett.}\\\/ {\\bf #1} (#2) #3}\n\\def\\PRT#1#2#3{{\\it Phys.\\ Rep.}\\\/ {\\bf#1} (#2) #3}\n\\def\\MODA#1#2#3{{\\it Mod.\\ Phys.\\ Lett.}\\\/ {\\bf A#1} (#2) #3}\n\\def\\IJMP#1#2#3{{\\it Int.\\ J.\\ Mod.\\ Phys.}\\\/ {\\bf A#1} (#2) #3}\n\\def\\nuvc#1#2#3{{\\it Nuovo Cimento}\\\/ {\\bf #1A} (#2) #3}\n\\def\\RPP#1#2#3{{\\it Rept.\\ Prog.\\ Phys.}\\\/ {\\bf #1} (#2) #3}\n\\def\\EJP#1#2#3{{\\it Eur.\\ Phys.\\ Jour.}\\\/ {\\bf C#1} (#2) #3}\n\\def{\\it et al\\\/}{{\\it et al\\\/}}\n\n\n\\hyphenation{su-per-sym-met-ric non-su-per-sym-met-ric}\n\\hyphenation{space-time-super-sym-met-ric}\n\\hyphenation{mod-u-lar mod-u-lar--in-var-i-ant}\n\n\n\\setcounter{footnote}{0}\n\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{Introduction}\n\nThe Standard Model of particle physics remains unscathed by contemporary\nexperiments. Its augmentation with the right--handed neutrinos, as envisioned\nby Pati and Salam nearly four decades ago\n\\cite{ps}, is mandated by solar and terrestrial\nneutrino observations. The Pati--Salam model naturally leads to the\nembedding of the standard model in $SO(10)$ representations. Most\nstrikingly the matter embedding in three 16 spinorial representations\ncorrelates the 54 gauge charges of the Standard Model states into\nthe single number of spinorial multiplets. The reduction\nin the number of experimental parameters from fifty four to one\nprovides the most important clue for the fundamental origins of the\nStandard Model. The remaining parameters, and in particular the\nflavour parameters, must find their origin in a theory that\nunifies gauge theories with gravity. It is then of further appeal\nthat heterotic--string theory accommodates\nthe $SO(10)$ embedding of the Standard Model matter spectrum.\nThree generation Heterotic--string models that preserve the\n$SO(10)$ embedding of the Standard Model states were constructed\nsince the late eighties \\cite{revamp,fny,alr,eu,cfn,lrs}.\n\nAbsence of higher order Higgs representations in heterotic--string models\nthat are based on level one Kac--Moody current algebras necessitates\nthat the $SO(10)$\nsymmetry is broken directly at the string level by discrete Wilson lines.\nA well known theorem due to Schellekens \\cite{schellekens} states that\nany such string model that preserves the\ncanonical $SO(10)$--GUT embedding\nof the weak hypercharge, and in which the non--Abelian GUT symmetries are\nbroken by discrete Wilson lines, necessarily contain states that carry\ncharges that do not obey the original GUT quantisation rule\n\\cite{schellekens}\\footnote{\nA similar observation was made in the context of Calabi--Yau compactification\nmodels with $E_6$ gauge group broken by Wilson lines \\cite{ww}.}.\nIn terms of the Standard Model charges these exotic states carry\nfractional electric charge. Electric charge conservation implies that the\nlightest of these states is stable, and their existence in nature\nis severely constrained by experiments \\cite{halyo}.\n\nWhile the existence of fractionally charged states in string models that\npreserve the canonical $SO(10)$ embedding of the Standard Model\nstates, and in which the $SO(10)$ symmetry is broken by Wilson lines,\nis mandated by Schellekens theorem, they may appear only in vector--like\nrepresentations, rather than in chiral representations. Superpotential\nterms for the vector--like states can then generate an intermediate\nor string scale\nmass to the exotic states, through the VEVs of Standard Model singlet fields\n\\cite{fc,cfn}.\nHowever, as the generation of the VEVs is obtained in an effective field theory\nanalysis a more appealing solution is to find string models in which the\nexotic fractionally charged states are confined to the massive\nspectrum. Recently, we demonstrated the existence of Pati--Salam vacua\nin which exotic fractionally charged\nstates do not exist in the massless spectrum \\cite{acfkr}. We dubbed\nsuch models as exophobic string vacua. We further showed that\nthere exist such exophobic Pati--Salam string models that contain\nthree generations and the required Higgs states to produce\nrealistic mass spectrum.\nWe demonstrated the existence of exophobic string vacua by utilising the\nfree fermionic classification techniques. These methods were developed in ref.\n\\cite{gkr} for type II string $N=2$ supersymmetric vacua.\nThey were extended in refs.\n\\cite{fknr,fkr} for the classification of heterotic $Z_2\\times Z_2$ free\nfermionic orbifolds, with unbroken $SO(10)$ and $E_6$ GUT symmetries,\nand in ref. \\cite{acfkr} heterotic--string vacua in which the $SO(10)$ symmetry\nis broken to the Pati--Salam subgroup.\n\nThe classification method used in\nrefs. \\cite{gkr,fknr, fkr, acfkr} utilises symmetric boundary conditions\nfor the set of internal world--sheet fermions that correspond to the\nsix dimensional compactified lattice. The symmetric boundary conditions\ncorrespond to $Z_2$ shifts in the compactified six dimensional torus\nand enable the scan of large sets of vacua. Such symmetric assignments in\nPati--Salam heterotic string models lead to the projection of the\nuntwisted Higgs bi--doublets and preservation of the corresponding\ncolour triplets \\cite{dts}. In quasi--realistic free fermionic models\nuntwisted Higgs doublets couple to twisted matter states.\nThe leading coupling is identified with the top quark mass\nterm in the superpotential \\cite{topyuk}.\nHence, this coupling is not present\nin the exophobic Pati--Salam models of ref. \\cite{acfkr}.\nThe question arises whether a top quark mass term exists\nin these string vacua. A viable\ntop quark Yukawa term is one of the first criteria\nthat a realistic string vacuum should admit.\n\nAn alternative to the twisted--twisted--untwisted coupling that\nis used in the quasi--realistic free fermionic models is\na twisted--twisted--twisted coupling. The existence of\na viable coupling is model dependent. The three states appearing\nin the trilevel term must arise from the three distinct twisted\nsectors. Hence, for example, if all the vectorial and spinorial\ntwisted states would arise from a single sector, the vacuum would\nnot be viable. In this paper we examine this question in the\nexophobic string vacuum of ref. \\cite{acfkr}. We show\nin one concrete model that the required coupling does\nexist. Additionally, we calculate the entire cubic level superpotential\nand show the existence of flat directions that leave a light pair\nof electroweak Higgs doublets and give heavy mass to all vector--like\ncolour triplets. Hence, below the Pati--Salam breaking scale\nthe spectrum of our model coincides with that of the Minimal\nSupersymmetric Standard Model (MSSM).\n\n\n\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{Exophobic Pati--Salam Heterotic--String Model}\\label{model}\n\nOur exophobic Pati--Salam heterotic--string model is constructed in the\nfree fermionic formulation \\cite{fff}.\nIn this formulation a string model is\nspecified in terms of a set of boundary condition basis vectors\n$v_i,i=1,\\dots,N$\n$$v_i=\\left\\{\\alpha_i(f_1),\\alpha_i(f_{2}),\\alpha_i(f_{3}))\\dots\\right\\},$$\nfor the 64 world--sheet real fermions \\cite{fff},\nand the one--loop Generalised GGSO projection coefficients,\n$ \\cc{v_i}{v_j}.$\nThe basis vectors span a space $\\Xi$ which consists of $2^N$ sectors that give\nrise to the string spectrum. Each sector, $\\eta\\in \\Xi$, is given by\n\\begin{equation}\n\\eta = \\sum N_i v_i,\\ \\  N_i =0,1\n\\end{equation}\nThe spectrum is truncated by a generalised GSO projection whose action on a\nstring state  $|S>$ is\n\\begin{equation}\\label{eq:gso}\ne^{i\\pi v_i\\cdot F_S} |S> = \\delta_{S}\\ \\cc{S}{v_i} |S>,\n\\end{equation}\nwhere $F_S$ is the fermion number operator and $\\delta_{S}=\\pm1$ is the\nspace--time spin statistics index.\nThe world--sheet free fermions in the light-cone gauge in the\nusual notation are:\n$\\psi^\\mu, \\chi^i,y^i, \\omega^i, i=1,\\dots,6$ (left-movers) and\n$\\bar{y}^i,\\bar{\\omega}^i, i=1,\\dots,6$,\n$\\psi^A, A=1,\\dots,5$, $\\bar{\\eta}^B, B=1,2,3$, $\\bar{\\phi}^\\alpha,\n\\alpha=1,\\ldots,8$ (right-movers).\nThe exophobic Pati--Salam model is\ngenerated by a set of thirteen basis vectors\n$\nB=\\{v_1,v_2,\\dots,v_{13}\\},\n$\nwhere\n\\begin{eqnarray}\nv_1=1&=&\\{\\psi^\\mu,\\\n\\chi^{1,\\dots,6},y^{1,\\dots,6}, \\omega^{1,\\dots,6}| \\nonumber\\\\\n& & ~~~\\bar{y}^{1,\\dots,6},\\bar{\\omega}^{1,\\dots,6},\n\\bar{\\eta}^{1,2,3},\n\\bar{\\psi}^{1,\\dots,5},\\bar{\\phi}^{1,\\dots,8}\\},\\nonumber\\\\\nv_2=S&=&\\{\\psi^\\mu,\\chi^{1,\\dots,6}\\},\\nonumber\\\\\nv_{2+i}=e_i&=&\\{y^{i},\\omega^{i}|\\bar{y}^i,\\bar{\\omega}^i\\}, \\\ni=1,\\dots,6,\\nonumber\\\\\nv_{9}=b_1&=&\\{\\chi^{34},\\chi^{56},y^{34},y^{56}|\\bar{y}^{34},\n\\bar{y}^{56},\\bar{\\eta}^1,\\bar{\\psi}^{1,\\dots,5}\\},\\label{basis}\\\\\nv_{10}=b_2&=&\\{\\chi^{12},\\chi^{56},y^{12},y^{56}|\\bar{y}^{12},\n\\bar{y}^{56},\\bar{\\eta}^2,\\bar{\\psi}^{1,\\dots,5}\\},\\nonumber\\\\\nv_{11}=z_1&=&\\{\\bar{\\phi}^{1,\\dots,4}\\},\\nonumber\\\\\nv_{12}=z_2&=&\\{\\bar{\\phi}^{5,\\dots,8}\\},\\nonumber\\\\\nv_{13}=\\alpha &=& \\{\\bar{\\psi}^{4,5},\\bar{\\phi}^{1,2}\\}.\\nonumber\n\\end{eqnarray}\nThe first two basis vectors generate a model with $N=4$\nspace--time supersymmetry and $SO(44)$ gauge group in four\ndimensions. The next six basis vectors correspond to freely\nacting shifts on the internal six dimensional compactified torus\nand reduce the gauge symmetry to $SO(32)$. The basis vectors $z_1$\nand $z_2$ are freely acting as well, and reduce the gauge symmetry\narising from the Neveu--Schwarz (NS) sector to\n$SO(16)\\times SO(8)\\times SO(8)$. Additional space--times\nvector bosons may arise from the sectors \\cite{fknr,fkr,acfkr}\n\\begin{equation}\n\\mathbf{G} =\n\\left\\{ \\begin{array}{cccccc}\nz_1          ,&\nz_2          ,&\n\\alpha       ,&\n\\alpha + z_1 ,&\n              &\n                \\cr\nx            ,&\nz_1 + z_2    ,&\n\\alpha + z_2 ,&\n\\alpha + z_1 + z_2,&\n\\alpha + x ,&\n\\alpha + x + z_1\n\\end{array} \\right\\} \\label{stvsectors}\n\\end{equation}\nand enhance the four dimensional gauge group. In (\\ref{stvsectors})\nwe defined the vector combination $$x=1+S+\\sum_{i=1}^6 e_i+z_1+z_2,$$\nwhich may enhance the observable $SO(16)$ gauge symmetry to $E_8$.\nFor suitable choices of the GGSO projection coefficients all\nthe space--time vector bosons arising from the sectors in eq.\n(\\ref{stvsectors}) are projected out.\nThe basis vectors $b_1$ and $b_2$ correspond to the\n$Z_2\\times Z_2$ twists of a $Z_2\\times Z_2$ orbifold.\nEach $Z_2$ twist reduces the number of supersymmetry generators\nfrom $N=4$ to $N=2$. In combination $b_1$ and $b_2$\nbreak $N=4$ to $N=1$ space--time supersymmetry,\nand reduce the NS gauge symmetry to\n$SO(10)\\times U(1)^3\\times SO(8)\\times SO(8)$.\n\nIn the quasi--realistic heterotic string models the gauge symmetries\nare realised as level one Kac--Moody algebras. The\nmassless spectrum of such models\ndoes not contain scalar Higgs multiplets in the adjoint\nrepresentation that can be used to break the non--Abelian $SO(10)$\nGUT symmetry. Consequently, the GUT gauge group must be broken\nat the string level, by a boundary condition basis vector in the\nfree fermionic formalism, or a discrete Wilson line in the orbifold\nformalism. The basis vector $\\alpha$ reduces the $SO(10)$ symmetry to\nthe Pati--Salam subgroup. The gauge group in our model is therefore:\n\\begin{eqnarray}\n{\\rm observable} ~: &~~~~~~~~SO(6)\\times SO(4) \\times U(1)^3 \\nonumber\\\\\n{\\rm hidden}     ~: &~~SO(4)^2\\times SO(8)~~~~             \\nonumber\n\\end{eqnarray}\nThe matter states in our model are embedded in $SU(4)\\times{SU(2)}_L\\times{SU(2)}_R$\nrepresentations as follows:\n\\begin{align}\n    {F}_L\\left({\\bf4},{\\bf2},{\\bf1}\\right)     &\\rightarrow\n     q\\left({\\bf3},{\\bf2},-\\frac 16\\right) + \\ell{\\left({\\bf1},{\\bf2},\\frac 12\\right)}\n     \\nonumber\\\\\n\\bar{F}_R\\left({\\bf\\bar 4},{\\bf1},{\\bf2}\\right)&\\rightarrow   u^c\\left({\\bf\\bar 3},\n{\\bf1},\\frac 23\\right)+d^c\\left({\\bf\\bar 3},{\\bf1},-\\frac 13\\right)+\n                            e^c\\left({\\bf1},\n                           {\\bf1},-1)+\\nu^c({\\bf1},{\\bf1},0\\right)\n                           \\nonumber\\\\\nh({\\bf1},{\\bf2},{\\bf2})\n &\\rightarrow   h^d\\left({\\bf1},{\\bf2},\\frac 12\\right) + h^u\\left({\\bf1},{\\bf2},-\\frac 12\\right)\n \\nonumber\\\\\nD\\left({\\bf6},{\\bf1},1\\right)  &\\rightarrow   d_3\\left({\\bf3},{\\bf1},\\frac 13\\right) +\n\\bar{d}_3\\left({\\bf\\bar 3},{\\bf1},-\\frac 13\\right),\n                             \\nonumber\n\\end{align}\nwhere $F_L$ and ${\\bar F}_R$ contain a single Standard Model generation;\n$h^d$ and $h^u$ are electroweak Higgs doublets; and $D$ contains\nvector--like colour triplets. The decomposition of the\nPati--Salam breaking Higgs fields in terms of the Standard Model group factors\nis:\n\\begin{align}\n\\bar{H}({\\bf\\bar 4},{\\bf1},{\\bf2})&\\rightarrow   u^c_H\\left({\\bf\\bar 3},\n{\\bf1},\\frac 23\\right)+d^c_H\\left({\\bf\\bar 3},{\\bf1},-\\frac 13\\right)+\n                            \\nu^c_H\\left({\\bf1},{\\bf1},0\\right)+\n                             e^c_H\\left({\\bf1},{\\bf1},-1\\right)\n                             \\nonumber\\\\\n{H}\\left({\\bf4},{\\bf1},{\\bf2}\\right)&\\rightarrow  u_H\\left({\\bf3},{\\bf1},-\\frac\n23\\right)+d_H\\left({\\bf3},{\\bf1},\\frac 13\\right)+\n              \\nu_H\\left({\\bf1},{\\bf1},0\\right)+ e_H\\left({\\bf1},{\\bf1},1\\right)\\nonumber\n\\end{align}\nThe electric charge in the Pati--Salam models is given by:\n\\begin{equation}\nQ_{em} = {1\\over\\sqrt{6}}T_{15}+{1\\over2}I_{3_L}+{1\\over2}I_{3_R}\n\\end{equation}\nwhere $T_{15}$ is the diagonal generator of $SU(4)$ and\n$I_{3_L}$, $I_{3_R}$\nare the diagonal generators of $SU(2)_L$, $SU(2)_R$, respectively.\n\nThe second ingredient that is needed to define the string vacuum\nare the GGSO projection coefficients that appear in the\none--loop partition function,\n$\\cc{v_i}{v_j}$, spanning a $13\\times 13$ matrix.\nOnly the elements with $i>j$ are\nindependent, and the others are fixed by modular invariance.\nA priori there are therefore 78 independent coefficients corresponding\nto $2^{78}$ distinct string vacua. Eleven coefficients\nare fixed by requiring that the models possess $N=1$ supersymmetry.\nAdditionally, imposing the\ncondition that the only space--time vector bosons that\nremain in the spectrum are those\nthat arise from the untwisted sector restricts the number of phases\nto a total of 51 independent GGSO phases. Each distinct configuration\nof these phases corresponds to a distinct vacuum. Some degeneracy in this\nspace of models may still exist due to additional symmetries over the entire\nspace. This is not relevant for our purposes here as our aim in this\nwork is to extract from the total space an exemplary model with the\nrequired phenomenological properties. Statistical analysis over the\nentire space was presented in ref. \\cite{acfkr}.\n\nThe breaking of the $SO(10)$ GUT symmetry by the $\\alpha$ boundary\ncondition basis vector results in combinations of the basis vectors\nthat can produce a priori massless states with fractional electric\ncharge. All these sectors, and the type of states that they a priori\ncan give rise to, are enumerated in ref. \\cite{acfkr}.\n\nBy employing an algorithm\nto generate random selection of the GGSO projection coefficient\nthe Pati--Salam free fermionic heterotic--string vacua were\nclassified in ref. \\cite{acfkr}.\nFor suitable choices of the GGSO projection coefficients\nall the massless\nfractionally charged states are projected out.\nFractionally charged states\nin this case only exist in the massive string spectrum, which is\ncompatible with experimental constraints.\nAn explicit choice of GGSO projection coefficients that produces a\nmodel with this property is given by:\n\n\\begin{equation} \\label{BigMatrix}  (v_i|v_j)\\ \\ =\\ \\ \\bordermatrix{\n      & 1& S&e_1&e_2&e_3&e_4&e_5&e_6&b_1&b_2&z_1&z_2&\\alpha\\cr\n 1    & 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 0\\cr\nS     & 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\cr\ne_1   & 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\cr\ne_2   & 1& 1& 0& 0& 1& 0& 0& 1& 0& 0& 1& 1& 0\\cr\ne_3   & 1& 1& 0& 1& 0& 1& 1& 0& 0& 0& 1& 0& 0\\cr\ne_4   & 1& 1& 0& 0& 1& 0& 1& 0& 0& 0& 1& 0& 1\\cr\ne_5   & 1& 1& 0& 0& 1& 1& 0& 0& 1& 0& 1& 1& 1\\cr\ne_6   & 1& 1& 0& 1& 0& 0& 0& 0& 0& 1& 1& 0& 0\\cr\nb_1   & 1& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 1& 0\\cr\nb_2   & 1& 0& 0& 0& 0& 0& 0& 1& 0& 1& 1& 0& 0\\cr\nz_1   & 1& 1& 0& 1& 1& 1& 1& 1& 0& 1& 1& 1& 1\\cr\nz_2   & 1& 1& 0& 1& 0& 0& 1& 0& 1& 0& 1& 1& 1\\cr\n\\alpha& 0& 1& 1& 0& 0& 1& 1& 0& 1& 1& 0& 1& 0\\cr\n  }\n\\end{equation}\nwhere we introduced the notation\n$\\cc{v_i}{v_j} = e^{i\\pi (v_i|v_j)}$.\n\n\\begin{table}[!h]\n\\noindent\n{\\small\n\\openup\\jot\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nsector&field&$SU(4)\\times{SU(2)}_L\\times{SU(2)}_R$&${U(1)}_1$&${U(1)}_2$&${U(1)}_3$\\\\\n\\hline\n$S$&$D_1$&$(6,1,1)$&$+1$&$\\hphantom{+}0$&$\\hphantom{+}0$\\\\\n&$D_2$&$(6,1,1)$&$\\hphantom{+}0$&$+1$&$\\hphantom{+}0$\\\\\n&$D_3$&$(6,1,1)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$+1$\\\\\n&$\\bar{D}_1$&$(6,1,1)$&$-1$&$\\hphantom{+}0$&$\\hphantom{+}0$\\\\\n&$\\bar{D}_2$&$(6,1,1)$&$\\hphantom{+}0$&$-1$&$\\hphantom{+}0$\\\\\n&$\\bar{D}_3$&$(6,1,1)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$-1$\\\\\n&$\\Phi_{12}$&$(1,1,1)$&$+1$&$+1$&$\\hphantom{+}0$\\\\\n&$\\Phi_{12}^{-}$&$(1,1,1)$&$+1$&$-1$&$\\hphantom{+}0$\\\\\n&$\\bar{\\Phi}_{12}$&$(1,1,1)$&$-1$&$-1$&$\\hphantom{+}0$\\\\\n&$\\bar{\\Phi}_{12}^{-}$&$(1,1,1)$&$-1$&$+1$&$\\hphantom{+}0$\\\\\n&$\\Phi_{13}$&$(1,1,1)$&$+1$&$\\hphantom{+}0$&$+1$\\\\\n&$\\Phi_{13}^-$&$(1,1,1)$&$+1$&$\\hphantom{+}0$&$-1$\\\\\n&$\\bar{\\Phi}_{13}$&$(1,1,1)$&$-1$&$\\hphantom{+}0$&$-1$\\\\\n&$\\bar{\\Phi}_{13}^-$&$(1,1,1)$&$-1$&$\\hphantom{+}0$&$+1$\\\\\n&$\\Phi_i,i=1,\\dots,6$&$(1,1,1)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$\\hphantom{+}0$\\\\\n&$\\Phi_{23}$&$(1,1,1)$&$\\hphantom{+}0$&$+1$&$+1$\\\\\n&$\\Phi_{23}^-$&$(1,1,1)$&$\\hphantom{+}0$&$+1$&$-1$\\\\\n&$\\bar{\\Phi}_{23}$&$(1,1,1)$&$\\hphantom{+}0$&$-1$&$-1$\\\\\n&$\\bar{\\Phi}_{23}^-$&$(1,1,1)$&$\\hphantom{+}0$&$-1$&$+1$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\\label{tablea}\\it\nUntwisted matter spectrum and\n$SU(4)\\times{SU(2)}_L\\times{SU(2)}_R\\times{U(1)}^3$ quantum numbers. }\n\\end{table}\n\nThe twisted massless states generated in the string vacuum\nof eq. (\\ref{BigMatrix}) produce the needed spectrum for viable\nphenomenology.\nIt contains three chiral generations; one pair of heavy Higgs\nstates to break the Pati--Salam gauge symmetry along a flat direction;\nlight Higgs bi-doublets needed to break the electroweak\nsymmetry and generate fermion masses; one vector sextet of $SO(6)$ needed\nfor the missing partner mechanism; it is completely free of massless\nexotic fractionally charged states.\nStates in vectorial representation are obtained in the free fermionic\nmodels from the untwisted Neveu--Schwarz sector and from twisted sectors that\ncontain four periodic world--sheet right--moving complex fermions.\nMassless states\nare obtained in such sectors by acting on the vacuum with a Neveu--Schwarz\nright--moving fermionic oscillator.\nThe model of eq. (\\ref{BigMatrix})\ncontains three pairs of untwisted $SO(6)$ sextets, and an additional\nsextet from a twisted sector.\nThese can obtain string scale mass along flat directions.\nAdditionally, it contains a number of $SO(10)$ singlet states,\nsome of which transform in non--trivial representations of the\nhidden sector gauge group.\nThe full massless spectrum of the model is shown in tables\n\\ref{tablea}, \\ref{tableb} and \\ref{tablec}, where we define\nthe vector combination $b_3\\equiv b_1+b_2+x$.\n\n\\begin{table}[!h]\n\\noindent\n{\\small\n\\openup\\jot\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nsector&field&$SU(4)\\times{SU(2)}_L\\times{SU(2)}_R$&${U(1)}_1$&${U(1)}_2$&${U(1)}_3$\\\\\n\\hline\n$S+b_2+e_1+e_6$&${F}_{1L}$&$({4},2,1)$&$\\hphantom{+}0$&$-{1\/2}$&$\\hphantom{+}0$\\\\\n$S+b_2+e_6$&$\\bar{F}_{1R}$&$(\\bar{4},1,2)$&$\\hphantom{+}0$&$-{1\/2}$&$\\hphantom{+}0$\\\\\n$S+b_3+e_1+e_2+e_3$&${F}_{2L}$&$({4},2,1)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n$S+b_1+e_4+e_5$&$\\bar{F}_{2R}$&$(\\bar{4},1,2)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}0$\\\\\n$S+b_1+e_3+e_4+e_5+e_6$&${F}_{1R}$&$({4},1,2)$&$-{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}0$\\\\\n$S+b_3+e_1+e_2+e_4$&$\\bar{F}_{3R}$&$(\\bar{4},1,2)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$\\hphantom{-}{1\/2}$\\\\\n$S+b_3+e_2+e_4$&${F}_{3L}$&$({4},2,1)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\n$S+b_3+e_2+e_3$&$\\bar{F}_{4R}$&$(\\bar{4},1,2)$&$\\hphantom{+}0$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n\\hline\n$S+b_2+x+e_2+e_5$&$h_1$&$(1,2,2)$&$-{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n$S+b_1+x+e_3+e_5$&$h_2$&$(1,2,2)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\n$S+b_1+x+e_3+e_5+e_6$&$h_3$&$(1,2,2)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\n\\hline\n$S+b_3+x+e_2$&$\\zeta_1$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\n$$&$\\bar{\\zeta}_1$&$(1,1,1)$&$-{1\/2}$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$\\\\\n\\hline\n$S+b_3+x+e_1+e_2+e_3+e_4$&$\\zeta_2$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$\\\\\n$ $&$\\bar{\\zeta}_2$&$(1,1,1)$&$-{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\n\\hline\n$S+b_2+x+e_1+e_2+e_5$&$D_4$&$(6,1,1)$&$-{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n&$\\zeta_a, a=3,4$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n&$\\bar{\\zeta}_a, a=3,4$&$(1,1,1)$&$-{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\n&$\\chi_+$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}1$\\\\\n&${\\chi}_-$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$&$-1$\\\\\n\\hline\n$S+b_1+x+e_3+e_4+e_5$&$      \\zeta_5$&$(1,1,1)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\n$                  $&$\\bar{\\zeta}_5$&$(1,1,1)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\n\\hline\n$S+b_1+x+e_4+e_5+e_6$&$      \\zeta_6$&$(1,1,1)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\n$                  $&$\\bar{\\zeta}_6$&$(1,1,1)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\n\\hline\n$S+b_2+x$&$      \\zeta_7$&$(1,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\n$                  $&$\\bar{\\zeta}_7$&$(1,1,1)$&$-{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\\label{tableb}\\it\nTwisted matter spectrum (observable sector)  and\n$SU(4)\\times{SU(2)}_L\\times{SU(2)}_R\\times{U(1)}^3$ quantum numbers. }\n\\end{table}\n\n\\begin{table}\n\\noindent\n{\\small\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nsector&field&${SU(2)}^4\\times{SO(8)}$&${U(1)}_1$&${U(1)}_2$&${U(1)}_3$\\\\\n\\hline\n$S+b_3+x+e_1+e_4$&$H_{12}^1$&$(2,2,1,1,1)$&$-{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_3+x+e_1+e_2+e_3$&$H_{12}^2$&$(2,2,1,1,1)$&$\\hphantom{+}{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_2+x+e_2+e_5+e_6$&$H_{12}^3$&$(2,1,2,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\\hline\n$S+b_3+x+e_2+e_3$&$H_{34}^1$&$(1,1,2,2,1)$&$\\hphantom{+}{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_3+x+e_1+e_2+e_4$&$H_{34}^2$&$(1,1,2,2,1)$&$-{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_2+x+e_1+e_2+e_5+e_6$&$H_{34}^3$&$(1,1,2,2,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\\hline\n$S+b_1+x+e_3+e_4+e_5+e_6$&$H_{34}^4$&$(1,1,2,2,1)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+e_4+e_5$&$H_{34}^5$&$(1,1,2,2,1)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\\hline\n$S+b_3+x+z_1$&$H_{13}^1$&$(2,1,2,1,1)$&$-{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_3+x+z_1+e_1+e_3+e_4$&$H_{13}^2$&$(2,1,2,1,1)$&$-{1\/2}$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_2+x+z_1+e_2$&$H_{13}^3$&$(2,1,2,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_2+x+z_1+e_2+e_6$&$H_{14}^1$&$(2,1,1,2,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+z_1+e_3$&$H_{14}^2$&$(2,1,1,2,1)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+z_1+e_6$&$H_{14}^3$&$(2,1,1,2,1)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\\hline\n$S+b_3+x+z_1+e_3+e_4$&$H_{24}^1$&$(1,2,1,2,1)$&$-{1\/2}$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_3+x+z_1+e_1$&$H_{24}^2$&$(1,2,1,2,1)$&$-{1\/2}$&$-{1\/2}$&$\\hphantom{+}0$\\\\\\hline\n$S+b_2+x+z_1+e_1+e_2$&$H_{24}^3$&$(1,2,1,2,1)$&$-{1\/2}$&$\\hphantom{+}0$&$-{1\/2}$\\\\\\hline\n$S+b_1+x+z_1+e_3+e_4$&$H_{24}^4$&$(1,2,1,2,1)$&$\\hphantom{+}0$&$-{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+z_1+e_4+e_6$&$H_{24}^5$&$(1,2,1,2,1)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$-{1\/2}$\\\\\\hline\n$S+b_2+x+z_1+e_1+e_2+e_6$&$H_{23}^1$&$(1,2,2,1,1)$&$\\hphantom{+}{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_2+x+z_2+e_2+e_5+e_6$&$Z_1$&$(1,1,1,1,8_c)$&$-{1\/2}$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+z_2+e_3+e_4$&$Z_2$&$(1,1,1,1,8_s)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\\hline\n$S+b_1+x+z_2+e_3+e_5$&$Z_3$&$(1,1,1,1,8_c)$&$\\hphantom{+}0$&$-{1\/2}$&$\\hphantom{+}{1\/2}$\\\\\\hline\n$S+b_1+x+z_2+e_4+e_6$&$Z_4$&$(1,1,1,1,8_s)$&$\\hphantom{+}0$&$-{1\/2}$&$-{1\/2}$\\\\\\hline\n$S+b_1+x+e_5+e_6$&$Z_5$&$(1,1,1,1,8_c)$&$\\hphantom{+}0$&$\\hphantom{+}{1\/2}$&$-{1\/2}$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\\label{tablec}\\it Twisted matter spectrum (hidden sector)  and\n${SU(2)}^4\\times{SO(8)}\\times{U(1)}^3$ quantum numbers.}\n\\end{table}\n\n\n\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{The superpotential and the top quark Yukawa}\n\nUsing the methodology of ref.\n\\cite{kln} for the calculation of renormalisable and\nnonrenormalisable terms, we calculate the cubic level\nsuperpotential of our exophobic Pati--Salam string model.\nIn particular, we seek to extract models that produce a cubic\nlevel mass term for the heavy generation, but not for the lighter generations,\nwhich should arise from higher order nonrenormalisable terms.\nThese requirements impose additional non--trivial constraints on the\nviable string vacua. Many models\ndo not produce any coupling of the form ${\\bar F}_R F_L h$. Such models do not\nadmit viable phenomenology as the models should produce at least a top\nquark mass term at leading order.\nSimilarly, models that produce leading mass terms for two or more families\nare not viable. The model presented in\nref. \\cite{acfkr} produces the cubic level terms\n$({\\bar F}_{1R}F_{3L}+{\\bar F}_{4R}F_{2L})h_3$. In this model therefore\ntwo heavy families may be degenerate in mass.\nMore appealing are therefore models that produce only a\nsingle mass term at leading order.\nThe model produced by eq. (\\ref{BigMatrix}) is an example of such a model.\nThe trilevel superpotential is given by\n\\begin{eqnarray}\n&~&\n\\frac{W_{\\rm SM}}{g\\,\\sqrt{2}}=\\nonumber\\\\\n&~&\n\\bar{F}_{2R}F_{3L}h_1+\n\\left\\{h_1h_1\\Phi_{13} + h_2h_2\\Phi_{23} + h_3h_3\\bar{\\Phi}_{23} + h_1h_3\\zeta_1\\right\\} +\n\\left\\{D_1D_2\\bar{\\Phi}_{12} +\n \\right. \\nonumber\\\\\n& &\n\\bar{D_1}D_2\\Phi_{12}^{-} + D_1\\bar{D_2}\\bar{\\Phi}_{12}^{-} + \\bar{D_1}\\bar{D_2}\\Phi_{12}\n+D_1D_3\\bar{\\Phi}_{13} +\\bar{D_1}D_3\\Phi_{13}^{-} +D_1\\bar{D_3}\\bar{\\Phi}_{13}^{-}  +\n\\nonumber\\\\\n& &\n\\left.\n                                                \\bar{D_1}\\bar{D_3}\\Phi_{13}\n+D_2D_3\\bar{\\Phi}_{23} +\\bar{D_2}D_3\\Phi_{23}^{-} +D_2\\bar{D_3}\\bar{\\Phi}_{23}^{-}  +\n                                               \\bar{D_2}\\bar{D_3}\\Phi_{23}\n\\right\\} +\n\\nonumber\\\\\n&~&\n \\left\\{\nD_{1}F_{1R}F_{1R} + \\bar{D}_{1}\\bar{F}_{2R}\\bar{F}_{2R}+\nD_{2}(\\bar{F}_{1R}\\bar{F}_{1R} + F_{1L}F_{1L})+\nD_{3}(\\bar{F}_{4R}\\bar{F}_{4R} + F_{2L}F_{2L}) +\n\\right. \\nonumber\\\\\n&~& \\left.\n                       \\bar{D}_{3}(\\bar{F}_{3R}\\bar{F}_{3R} + F_{3L}F_{3L})+\nD_4 ( \\bar{F}_{2R}\\bar{F}_{3R} + D_2\\chi_- +  \\bar{D}_2\\chi_+ +\nD_4\\Phi_{13})\\right\\}+\\nonumber\\\\\n&~&\n\\bar{\\Phi }_{13} \\chi_-\\chi_++\\Phi _{23} \\bar{\\Phi}_{12} \\Phi _{13}^-\n+\\Phi _{13} \\bar{\\Phi }_{12} \\Phi_{23}^-+\\Phi _{23} \\bar{\\Phi }_{13} \\Phi _{12}^-\n+\\Phi_{12}^- \\Phi _{23}^- \\bar{\\Phi }_{13}^- +\\nonumber\\\\\n&~&\n\\Phi _{13}\n   \\bar{\\Phi }_{23} \\bar{\\Phi }_{12}^-+\\Phi _{12}\n   \\bar{\\Phi }_{23} \\bar{\\Phi }_{13}^-+\\Phi _{13}^-\n   \\bar{\\Phi }_{12}^- \\bar{\\Phi }_{23}^-+\\Phi _{12}\n   \\bar{\\Phi }_{13} \\bar{\\Phi }_{23}^- +\n \\nonumber\\\\\n&~&\n \\zeta_1{}^2 \\bar{\\Phi}_{12}^-+\\bar{\\zeta}_1{}^2 \\Phi_{12}^-+\n\\left(\\zeta_3{}^2+\\zeta_4{}^2+\\zeta_7{}^2\\right)\n   \\bar{\\Phi}_{13}^-+\\left(\\bar{\\zeta}_3{}^2+\\bar{\\zeta}_4{}^2+\n\\bar{\\zeta}_7{}^2\\right) \\Phi _{13}^- +\n    \\nonumber\\\\\n&~&\n\\frac{1}{2} \\bar{\\zeta}_2\n   \\bar{\\zeta}_5 \\chi_++\\zeta_2{}^2 \\bar{\\Phi }_{12}+\\left(\\zeta_5{}^2+\n\\zeta_6{}^2\\right) \\bar{\\Phi }_{23}+\\Phi _{12}\n   \\bar{\\zeta}_2{}^2+\\Phi _5 \\left(\\zeta_1 \\bar{\\zeta}_1+\\zeta_2\n   \\bar{\\zeta}_2\\right) +\n    \\nonumber\\\\\n&~&\n\\Phi _2 \\left(\\zeta_5 \\bar{\\zeta}_5+\\zeta_6\n   \\bar{\\zeta}_6\\right)+\\Phi _{23}\n   \\left(\\bar{\\zeta}_5{}^2+\\bar{\\zeta}_6{}^2\\right)+\\Phi _4 \\zeta_7\n   \\bar{\\zeta}_7+\\frac{\\zeta_4 \\zeta_5\n   \\bar{\\zeta}_2}{\\sqrt{2}}+\\frac{\\zeta_2 \\bar{\\zeta}_3\n   \\bar{\\zeta}_5}{\\sqrt{2}}\n\\end{eqnarray}\n\nThe string vacuum contains three anomalous $U(1)$s\n\\begin{equation}\n{\\rm Tr}{U(1)}_1=-12~~;~~\n{\\rm Tr}{U(1)}_2=-24~~;~~\n{\\rm Tr}{U(1)}_3=-12\n\\end{equation}\nredefining we obtain two anomaly-free\n\\begin{eqnarray}\n{U(1)}'_1={U(1)}_1-{U(1)}_3\\label{u1p}\\\\\n{U(1)}'_2={U(1)}_1-{U(1)}_2+{U(1)}_3\\label{u2p}\n\\end{eqnarray}\nand one anomalous combination\n\\begin{eqnarray}\n{U(1)}'_A={U(1)}_1+2\\,{U(1)}_2+{U(1)}_3\\ ,\\ {\\rm Tr}{U(1)}_A=-72\\label{upa}\n\\end{eqnarray}\n\nThe electroweak Higgs doublets come in pairs and are accommodated\nin the Pati--Salam bi-doublets $h_1,h_2,h_3$. Their mass matrix is\n\\begin{eqnarray}\nM_h\\sim\\bordermatrix{\n&h_1&h_2&h_3\\cr\nh_1&\\Phi_{13}&\\frac{\\zeta_1}{\\sqrt{2}}&0\\cr\nh_2&\\frac{\\zeta_1}{\\sqrt{2}}&\\bar{\\Phi}_{23}&0\\cr\nh_3&0&0&\\Phi_{23}}\n\\end{eqnarray}\nIn order to keep $h_1$ massless we need to impose the condition\n\\begin{equation}\n\\Phi_{13}\\,\\bar{\\Phi}_{23}-\\frac{\\zeta_1^2}{2}=0.\n\\label{doubletconstraint}\n\\end{equation}\n\nNext, we discuss the colour--triplet mass matrix in our string derived\nPati--Salam model. Three pairs of colour--triplets arise in the model from\nthe untwisted Neveu--Schwarz sector, and are accommodated in the sextet of the\nPati--Salam $SU(4)$. we denote these by\n$D_i=d_i(3,1,1)+d^c_i(\\bar{3},1,1)$,\n$\\bar{D}_i=\\bar{d}_i({\\bar 3},1,1)+\\bar{d}^c_i({3},1,1)$.\nAn additional sextet arises in the model from a twisted sector.\nA further pair of colour triplets is obtained from\nthe heavy Higgs states,\n$\\bar{F}_{1R}$ and $F_{1R}$ that are used to break the Pati--Salam\nsymmetry, and must get a VEV of the order of the GUT scale.\nWe denote the colour triplets in these fields by\n$F_{\\alpha R}=d_{\\alpha H}+\\dots$. At the cubic level the colour triplet mass\nmatrix then takes the form,\n\\begin{eqnarray}\nM_D=\\bordermatrix{\n&d_1&d_2&d_3&\\bar{d}_1&\\bar{d}_2&\\bar{d}_3&d_4&d_{1H}\\cr\nd^c_1&0 & \\bar{\\Phi }_{12} & \\bar{\\Phi }_{13} & 0\n& \\bar{\\Phi }_{12}^- & \\bar{\\Phi }_{13}^- & 0 & F_{1R}\\cr\nd^c_2&\\bar{\\Phi }_{12} & 0 & \\bar{\\Phi }_{23} & \\Phi _{12}^-\n& 0 & \\bar{\\Phi }_{23}^- & \\chi_- & 0 \\cr\nd^c_3&\\bar{\\Phi }_{13} & \\bar{\\Phi }_{23} & 0 & \\Phi _{13}\n& \\Phi _{23}^- & 0 & 0 & 0 \\cr\n\\bar{d}^c_1&0 & \\Phi _{12}^- & \\Phi _{13} & 0 & \\Phi _{12}\n& \\Phi _{13}  &0& 0 \\cr\n\\bar{d}^c_2&\\Phi _{12}^- & 0 & \\Phi _{23}^- & \\Phi _{12} & 0\n& \\Phi _{23} & \\chi_+ & 0 \\cr\n\\bar{d}^c_3&\\bar{\\Phi }_{13}^- & \\bar{\\Phi }_{23}^- & 0\n& \\Phi _{13} & \\Phi _{23} & 0 & 0 & 0 \\cr\nd^c_4&0 & \\chi_- & 0 & 0 & \\chi_+ & 0 & \\Phi _{13} & 0 \\cr\n\\bar{d}^c_{1H}&0 & \\bar{F}_{1R} & 0 & 0 & 0 & 0 & 0 & 0}\n\\label{tmm}\n\\end{eqnarray}\nWe have ${\\rm det}(M_D)\\sim \\Phi_{13}^2$ so in order to keep\ntriplets heavy and $h_1$ light we need\n$\\{\\Phi_{13}, \\zeta_1,\\bar{\\Phi}_{23}\\}\\ne0$.\n\nNext, we examine the pattern of symmetry breaking. The anomalous $U(1)_A$\nis broken by the Green--Schwarz--Dine--Seiberg--Witten mechanism \\cite{dsw}\nin which a potentially large Fayet--Iliopoulos $D$--term\n$\\xi$ is generated by the VEV of the dilaton field.\nSuch a $D$--term would, in general, break supersymmetry, unless\nthere is a direction $\\hat\\phi=\\sum\\alpha_i\\phi_i$ in the scalar\npotential for which $\\sum Q_A^i\\vert\\alpha_i\\vert^2<0$ and that\nis $D$--flat with respect to all the non--anomalous gauge symmetries\nalong with $F$--flat. If such a direction\nexists, it will acquire a VEV, cancelling the Fayet--Iliopoulos\n$\\xi$--term, restoring supersymmetry and stabilising the vacuum.\nAssuming VEVs for the non-Abelian gauge singlets and a pair of PS breaking Higgs,\n$F_{1R}=\\bar{F}_{1R}=M_G$,\nthe $D$--flatness constraints in our model are given by:\n\\begin{eqnarray}\n{U(1)}'_1 &:& \\left(\\left|\\Phi_{12}\\right|^2-\\left|\\bar{\\Phi}_{12}\n\\right|^2\\right)+\n\\left(\\left|\\Phi_{12}^{-}\\right|^2-\\left|\\bar{\\Phi}_{12}^{-}\\right|^2\\right)\n+2\\left(\\left|\\Phi_{13}^{-}\\right|^2-\\left|\\bar{\\Phi}_{13}^{-}\\right|^2\\right)\n\\nonumber\\\\\n&~&-\\left(\\left|\\Phi_{23}\\right|^2-\\left|\\bar{\\Phi}_{23}\\right|^2\\right)\n+\\left(\\left|\\Phi_{23}^-\\right|^2-\\left|\\bar{\\Phi}_{23}^-\\right|^2\\right)\n+\\frac{1}{2}\\sum_{i=1,2}\\left(\\left|\\zeta_{i}\\right|^2-\\left|\\bar{\\zeta}_{i}\\right|^2\\right)\n\\nonumber\\\\\n&~&\n-\\frac{1}{2}\\sum_{i=5,6}\\left(\\left|\\zeta_{i}\\right|^2-\\left|\\bar{\\zeta}_{i}\n\\right|^2\\right)\n+\\sum_{i=3,4,7}\\left(\\left|\\zeta_{i}\\right|^2-\\left|\\bar{\\zeta}_{i}\\right|^2\\right)-\\frac{1}{2}\\left|{F}_{1R}\\right|^2\n=0\\label{du1}\n\\end{eqnarray}\n\\begin{eqnarray}\n{U(1)}'_2 &:&\n2\\left(\\left|\\Phi_{12}^{-}\\right|^2-\\left|\\bar{\\Phi}_{12}^{-}\\right|^2\\right)\n+2\\left(\\left|\\Phi_{13}\\right|^2-\\left|\\bar{\\Phi}_{13}\\right|^2\\right)\n-2\\left(\\left|\\Phi_{23}^-\\right|^2-\\left|\\bar{\\Phi}_{23}^-\\right|^2\\right)\\nonumber\\\\\n&~&+\n\\left(\\left|\\zeta_{1}\\right|^2-\\left|\\bar{\\zeta}_{1}\\right|^2\\right)+\n2\\left|\\chi_{-}\\right|^2+\\frac{1}{2}\\left(\\left|\\bar{F}_{1R}\\right|^2-\\left|F_{1R}\\right|^2\\right)=0\\label{du2}\n\\\\\n{U(1)}'_A &:&\n3\\left(\\left|\\Phi_{12}\\right|^2-\\left|\\bar{\\Phi}_{12}\\right|^2\\right)-\n\\left(\\left|\\Phi_{12}^{-}\\right|^2-\\left|\\bar{\\Phi}_{12}^{-}\\right|^2\\right)\n+2\\left(\\left|\\Phi_{13}\\right|^2-\\left|\\bar{\\Phi}_{13}\\right|^2\\right)\n\\nonumber\\\\\n&~&+\n3\\left(\\left|\\Phi_{23}\\right|^2-\\left|\\bar{\\Phi}_{23}\\right|^2\\right)\n+\\left(\\left|\\Phi_{23}^-\\right|^2-\\left|\\bar{\\Phi}_{23}^-\\right|^2\\right)\n-\\frac{1}{2}\\left(\\left|\\zeta_{1}\\right|^2-\\left|\\bar{\\zeta}_{1}\\right|^2\\right)\n\\nonumber\\\\\n&~&\n+\\frac{3}{2}\\sum_{i=2,5,6}\\left(\\left|\\zeta_{i}\\right|^2-\n\\left|\\bar{\\zeta}_{i}\\right|^2\\right)\n+3\\left|\\chi_{+}\\right|^2-\\left|\\chi_{-}\\right|^2\n\\nonumber\\\\\n&~&\n-\\frac{1}{2}\\left|F_{1R}\\right|^2-\\left|\\bar{F}_{1R}\\right|^2=\n+\\frac{3\\,g^2}{16\\pi^2}M^2\\equiv\\xi. \\label{duA}\n\\end{eqnarray}\nIn eq. (\\ref{duA}) $g$ is the gauge coupling in the effective field theory,\nand $M$ is the so--called reduced Planck mass\n$M\\equiv M_{\\rm Planck}\/\\sqrt{8\\pi}$. In setting $\\xi$ we followed\nthe conventions of \\cite{cew}.\nThe set of $F$--flatness constraints are obtained by requiring\n\\begin{equation}\n\\langle F_i\\equiv\n{{\\partial W}\\over{\\partial\\eta_i}}\\rangle=0\\label{fterms}\n\\end{equation}\nwhere $\\eta_i$ are all the fields that appear in the model.\nThe solution ({\\it i.e.}\\  the choice of fields\nwith non--vanishing VEVs) to the set of\nequations (\\ref{du1})--(\\ref{fterms}),\nthough nontrivial, is not unique. Therefore in a typical model there exist\na moduli space of solutions to the $F$ and $D$ flatness constraints,\nwhich are supersymmetric and degenerate in energy \\cite{moduli}.\nAssuming VEVs for the non-Abelian gauge singlets and a pair of PS breaking Higgs,\n$F_{1R}=\\bar{F}_{1R}=M_G$, the following 9 parameter exact solution\n\\begin{equation}\n\\left\\{\\Phi _3,\\Phi _4,\\Phi _6,\\bar{\\Phi }_{23},\\Phi_{23}^-,\n\\bar{\\Phi }_{23}^-,\\Phi _{13}^-,\\bar{\\Phi}_{13}^-,\n\\bar{\\Phi }_{12}\\right\\}\n\\label{freeparameters}\n\\end{equation}\nsatisfies all $F$-flatness equations while keeping one linear\ncombination of the bi-doublets ($h_1,h_2$) massless:\n\\begin{eqnarray}\n0&=&\\Phi_1=\\Phi_2=\\chi_+=\\chi_-=\\zeta_i=\n\\bar{\\zeta}_i,i=3,\\dots,7\\label{fphi12zeta37}\\\\\n\\Phi_5&=&-\\frac{2i}{\\sqrt{3}}\\frac{\\bar{\\Phi}_{12}}{\\bar{\\Phi}_{23}}\n\\sqrt{\\frac{\\Phi_{13}^- \\Phi_{23}^- \\bar{\\Phi}_{23}^-}{\\bar{\\Phi}_{13}^-}}\n\\label{fphi5}\\\\\n\\Phi_{23}&=&\\frac{\\Phi_{23}^-\\bar{\\Phi}_{23}^-}{\\bar{\\Phi}_{23}} \n~~~~~~~~~~~~~~~~~~~~~~,~~~~\n\\Phi_{13} = -\\frac{\\Phi_{13}^- \\bar{\\Phi}_{23}^-}{3\\bar{\\Phi}_{23}}\\label{fphi13}\\\\\n\\bar{\\Phi}_{13}&=&-\\frac{3 \\bar{\\Phi }_{23}\n\\bar{\\Phi }_{13}^-}{\\bar{\\Phi }_{23}^-} ~~~~~~~~~~~~~~~~~~,~~~~\n\\Phi_{12} = -\\frac{\\bar{\\Phi }_{12} \\Phi _{13}^- \\Phi _{23}^- \\bar{\\Phi\n   }_{23}^-}{3 \\bar{\\Phi }_{23}{}^2 \\bar{\\Phi }_{13}^-}\\label{fphi12}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\Phi_{12}^-&=&\\frac{\\bar{\\Phi }_{12} \\Phi _{13}^- \\bar{\\Phi }_{23}^-}{3\n   \\bar{\\Phi }_{23} \\bar{\\Phi }_{13}^-} ~~~~~~~~~~~~~~~~~~,~~~~\n\\bar{\\Phi}_{12}^- = -\\frac{\\bar{\\Phi }_{12} \\Phi _{23}^-}{\\bar{\\Phi }_{23}}\\\\\n\\zeta_1&=&i \\sqrt{\\frac{2\\Phi _{13}^- \\bar{\\Phi }_{23}^-}{3}}  ~~~~~~~~~~~~~~,~~~~\n~\\bar{\\zeta}_1 = - \\sqrt{2 \\Phi _{23}^- \\bar{\\Phi }_{13}^-}\\\\\n\\zeta_2&=&i \\sqrt{\\frac{2\\Phi _{23}^-\n   \\Phi _{13}^- \\bar{\\Phi }_{23}^-}{3\\bar{\\Phi }_{23}}} ~~~~~~~~~~~~~,~~~~\n~\\bar{\\zeta}_2 = \\sqrt{2 \\bar{\\Phi }_{23} \\bar{\\Phi }_{13}^-}\\label{fzeta2}\n\\end{eqnarray}\n\nThe triplet mass matrix \\eqref{tmm} determinant is\n\\begin{eqnarray}\n\\det{M_D}=  -\\frac{64}{27}\\frac{F_{1R}\n\\bar{F}_{1R}\\bar{\\Phi }_{12} \\Phi _{13}^-{}^3\n   \\Phi _{23}^-{}^2 \\bar{\\Phi }_{23}^-{}^3}{\\bar{\\Phi}_{23}{}^3}\n\\end{eqnarray}\nand thus all triplets are massive.\n\nFor this $F$--flatness solution, the  three $D$--flatness equations (\\ref{du1}--\\ref{duA}) depend on seven\nparameters,\n$\\vert {\\bar\\Phi}_{23}\\vert,~\n\\vert {\\Phi}_{23}^-\\vert,~\n\\vert {\\bar\\Phi}_{23}^-\\vert,~\n\\vert {\\Phi}_{13}^-\\vert,~\n\\vert {\\bar\\Phi}_{13}^-\\vert,~\n\\vert {\\bar\\Phi}_{12}\\vert,$\nand $\\vert F_{1R}\\vert = \\vert {\\bar F}_{1R}\\vert.$\nSetting $\\vert F_{1R}\\vert = \\vert {\\bar F}_{1R}\\vert = M_{\\rm G}=0.02 \\sqrt{\\xi}$\nthe $D$--flatness equations can be solved numerically in terms of three\nparameters. Choosing, for example,\n$\\vert {\\bar\\Phi}_{23}\\vert=\\vert {\\bar\\Phi}_{13}^-\\vert =\n{1\\over2} \\vert {\\bar\\Phi}_{23}^- \\vert = \\chi$\nwe can solve numerically for\n$\\vert {\\Phi}_{13}^- \\vert$,  $\\vert {\\bar\\Phi}_{23}^- \\vert$\nand\n$\\vert {\\bar\\Phi}_{12}^- \\vert$.\nThe results are shown in figure \\ref{sold}.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=10cm]{solda.eps}\n\\caption{\\label{sold}\n\\it Solution of of the $D$--flatness equations for\n$\\vert {\\Phi}_{13}^- \\vert$,  $\\vert {\\bar\\Phi}_{23}^- \\vert$\nand\n$\\vert {\\bar\\Phi}_{12}^- \\vert$  as a\nfunction of $\\chi= \\vert {\\bar\\Phi}_{23}\\vert=\\vert {\\bar\\Phi}_{13}^-\\vert =\n{1\\over2} \\vert {\\bar\\Phi}_{23}^- \\vert$ (all VEVs are in units of  $\\sqrt{\\xi}$).}\n\\end{figure}\nIn figure \\ref{mdlight} we plot the mass of the two lightest  colour triplets for\nthe one parameter solution displayed in figure \\ref{sold}.\nFrom the figure we note that for singlet VEVs of the order of $0.1\\sqrt{\\xi}$\nthe lightest triplet mass is of the order of $0.4M_{\\rm GUT}$. Thus the additional colour\ntriplets are heavy enough to protect proton from decaying through dangerous  triplet mediated  dim-5\noperators \\cite{lpd}. Additionally, we note that the three $U(1)$ symmetries\nin eqs. (\\ref{u1p}, \\ref{u2p}, \\ref{upa}) are broken in the $F$-- and $D$--flat vacuum.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=12cm]{mdlighta.eps}\n\\caption{\\label{mdlight}\n\\it The ratio  of the two lightest colour triplet mass over $M_{\\rm GUT}$ as a\nfunction of $\\chi= \\vert {\\bar\\Phi}_{23}\\vert=\\vert {\\bar\\Phi}_{13}^-\\vert =\n{1\\over2} \\vert {\\bar\\Phi}_{23}^- \\vert$ (in units of  $\\sqrt{\\xi}$). }\n\\end{figure}\n\n\n\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{Conclusions}\\label{conclude}\n\nIn this paper we analysed the phenomenology of an exemplary exophobic\nPati--Salam heterotic string vacuum, in which exotic fractionally\ncharged states exist in the massive spectrum, but not among the\nmassless states. In that respect the exophobic models are distinguished\nfrom other models in which exotic states gain heavy mass\nby vacuum expectation values of Standard Model singlet fields.\nOur exophobic model also contains the\nHiggs representations that are needed to break the gauge symmetry\nto that of the Standard Model and to generate fermion masses\nat the electroweak scale. One can then start to probe the\nphenomenology of such models in more detail. We showed\nin particular that the presence of a top\nquark Yukawa coupling at leading order places\nan additional strong constraint on the viability of the models.\nIn many models a top quark Yukawa may not exist at all, whereas\nin others two or more generations may obtain a mass term at leading\norder. In our exemplary model a mass term at leading\norder exist only for one family. Additionally,\nwe demonstrated the existence of supersymmetric $F$-- and $D$--flat\ndirections that give heavy mass to all the\ncolour triplets beyond those of the Standard Model\nand leave one pair of electroweak Higgs doublets light.\nHence, below the Pati--Salam breaking scale the spectrum of our\nmodel consists solely of that of the Minimal Supersymmetric\nStandard Model. We remark that while there exist other models\nin which the exotic states are decoupled along flat directions,\nin many of these models the mass scale of the exotic states\nis ambiguous as the relevant mass terms arise from higher\norder superpotential terms that are expected to be suppressed\ncompared to the leading string scale mass terms \\cite{raby}.\nThe novelty in our model is that the exotic states are\nabsent from the massless spectrum to begin with and\nhence necessarily have string scale masses. In this respect the model\nis superior to earlier constructions. Further analysis of higher\norder terms in the superpotential can now be pursued to\nconfront the model with the detailed Standard Model mass\nand mixing data. We note that the interplay between statistical\nsearches and detailed analysis of specific models takes us a step further\ntoward the construction of string models that reproduce the phenomenological\nStandard Model data. We will return to these issues in future publications.\n\n\n\n\n\n\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus .2ex{Acknowledgements}\n\nAEF would like to thank the University of Oxford for hospitality.\nAEF is supported in part by STFC under contract PP\/D000416\/1.\nJR work is supported in part by the EU under contract\nPITN-GA-2009-237920.\n\n\n\n\n\\bigskip\n\\medskip\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Kuramoto model \\cite{Kuramoto1975Self} is a simple mathematical model\nthat describes the dynamics on networks of oscillators.\nIt has found applications in neuroscience, biology, chemistry and power systems  \n\\cite{generative2010breakspear,synchronization2010dorfler,GUO2021106804,RODRIGUES20161}.\nDespite its simplicity, it exhibits interesting emergent behaviors.\nOf particular interest is the phenomenon of frequency synchronization\nwhich is when oscillators spontaneously synchronize to a common frequency.\nIn a rotational frame,\na synchronization configuration is an equilibrium of the dynamical system,\ni.e., a root of the Kuramoto equations.\nThis paper aims to understand the structure of these roots\nin ``typical'' and ``atypical'' networks.\n\n\nEarlier studies focused on statistical analysis of\ninfinite networks \\cite{Kuramoto1975Self} but more\nrecently, tools from differential and algebraic geometry\nhave enabled analysis of synchronization on finite networks.\nFor a finite network, knowing the total number of synchronization configurations\nis fundamental to our understanding of this model.\nFrom a computational perspective,\nthis knowledge also plays a critical role in\ndeveloping numerical methods for finding synchronization configurations.\nFor instance, this number serves as a stopping criterion\nfor \\emph{monodromy} algorithms\n\\cite{lindberg2020exploiting}\nand allows for the development of specialized\n\\emph{homotopy} algorithms \\cite{Chen2019Directed,ChenDavis2022Toric}\nfor finding synchronization configurations.\n\n\nIn 1982, Ballieul and Byrnes introduced root counting techniques\nfrom algebraic geometry to this field and showed that\na Kuramoto network of $N$ oscillators  \nhas at most $\\binom{2N-2}{N-1}$ synchronization configurations \\cite{BaillieulByrnes1982Geometric}.\nAlgebraic geometers will recognize this bound as the bi-homogeneous B\\'ezout bound\nfor an algebraic version of the defining equations.\nThis upper bound can be reached when the network is complete\nand complex roots (i.e., complex relaxation of the configurations) are counted.\nHowever, for sparse networks, \nthe root count (even counting complex roots)\ncan be significantly lower than this upper bound \\cite{GuoSalam1990Determining,MolzahnMehtaNiemerg2016Toward},\ndemonstrating the need for a network-dependent root count.\n\nGuo and Salam initiated one of the first algebraic analyses\non such sparsity-dependent root counts \\cite{GuoSalam1990Determining}.\nMolzahn, Mehta, and Niemerg provided computational evidence\nfor the connection between this root count\nand network topology \\cite{MolzahnMehtaNiemerg2016Toward}.\nIn the special case of rank-one coupling,\nCoss, Hauenstein, Hong and Molzahn\nproved this complex root count to be $2^N - 2$,\nwhich is also an asymptotically sharp bound on the real root count\n\\cite{coss2018locating}.\nChen, Davis and Mehta established the maximum complex root count\nfor cycle networks to be $N \\binom{N-1}{ \\lfloor (N-1) \/ 2 \\rfloor }$ \\cite{ChenDavisMehta2018Counting},\nwhich is sharp as it can be reached\nwith generic choices of coefficients,\nand it is asymptotically smaller than the bi-homogeneous B\\'ezout bound\ndiscovered by Baillieul and Byrnes,\nproving that sparse networks have significantly fewer synchronization configurations than dense ones. \nInterestingly, Lindberg, Zachariah, Boston, and Lesieutre showed that\nthis bound is attainable by real roots\n\\cite{LindbergZachariahBostonLesieutre2022Distribution}.\nThis bound is an instance of the \n``adjacency polytope bound'' \\cite{Chen2019Unmixing}.\nHowever, the following was still unknown. \n\n\n\\begin{question}\\label{q1}\n    For generic choices of coupling coefficients and natural frequencies,\n    will the complex root count for the algebraic Kuramoto equations\n    reach the adjacency polytope bound for all networks?\n    If true, this system is Bernshtein-general,\n    despite the constraints on the coefficients.\n\\end{question}\nIn the first part of this paper,\nwe provide a positive answer to this question\nand thus establish the generic root count for the Kuramoto equations\nderived from a graph $G$\nto be the normalized volume of the adjacency polytope\n$\\conv \\{ \\colv{e}_i - \\colv{e}_j \\}_{ \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G) }$.\nWe note that this result is similar to recent work in \\cite{breiding2022the} which shows that the number of (approximate) complex solutions to the Duffing equations is generically the volume of the \\emph{Oscillator polytope}.\nWe also extend this generic root count result to\nvariations of the Kuramoto equations.\n\n\nEven though the generic behavior holds for almost all parameters,\nunderstanding the two types of non-generic behavior is still of interest.\nFirst, the roots of the Kuramoto equations may remain isolated\nbut the total number can drop below the generic root count.\nThe second part of this paper considers this situation and aims to answer the following.\n\\begin{question}\\label{q2}\n    What are the conditions on the coupling coefficients\n    under which the complex roots count for Kuramoto equations\n    will drop below the generic root count?\n\\end{question}\nFurthermore, for certain networks, we answer the following.\n\\begin{question}\\label{q3}\n    What is the gap between\n    the generic root count and actual root count?\n\\end{question}\nFor the second type of non-generic behavior,\nnon-isolated solutions may appear.\nAshwin, Bick, and Burylko analyzed certain types of\nnon-isolated solutions in complete networks of identical oscillators\n\\cite{AshwinBickBurylko2016Identical}.\nA concrete example of this in a network of four identical oscillators\nwith uniform coupling coefficients was described in \\cite[Example 2.1]{coss2018locating}.\nThe existence of non-isolated equilibria for cycle networks\nwas discovered by Lindberg, Zachariah, Boston and Lesieutre \n\\cite{LindbergZachariahBostonLesieutre2022Distribution}.\nRecent work by Sclosa shows that for every $d \\geq 1$\nthere is a Kuramoto network whose stable equilibria\nform a manifold of dimension $d$ \\cite{sclosa2022kuramoto}.\nIn the last part, we aim to answer the following.\n\n\n\\begin{question}\\label{q4}\n    What are the conditions on the coupling coefficients\n    under which there exist non-isolated synchronization configurations?\n   What are these configurations?\n\\end{question}\n\n\nThe rest of this paper is structured as follows.\nIn \\Cref{sec: prelims}, we review concepts\nthat will be used throughout this paper.\nWe then consider the generic root count of the Kuramoto equations in \\Cref{sec: generic root count} and provide an answer to \\Cref{q1}.\nNext, we turn our attention to non-generic coupling coefficients.\nIn \\Cref{sec: explicit genericity conditions} we consider when all solutions are isolated and the number of complex solutions drops, answering \\Cref{q2,q3}.\nIn \\Cref{sec: +dimensional} we identify parameter values under which the Kuramoto equations have positive dimensional components, answering \\Cref{q4}.\nFinally, we conclude with a few remarks in \\Cref{sec: conclusion}.\n\n\n\n\\section{Notation and preliminaries}\\label{sec: prelims}\n\nColumn vectors, representing points of the lattice $L \\cong \\ensuremath{\\mathbb{Z}}^n$,\nare denoted by lowercase letters with arrowhead, e.g., $\\colv{a}$.\nWe use boldface letters, e.g., $\\rowv{x}$,\nfor points in $\\ensuremath{\\mathbb{C}}^n$, $\\ensuremath{\\mathbb{R}}^n$, or the dual lattice $L^\\vee \\cong \\ensuremath{\\mathbb{Z}}^n$,\nand they are written as row vectors.\nFor $\\rowv{x} = (x_1, \\dots, x_n)$\nand $\\colv{a} = (a_1,\\dots,a_n)^\\top \\in \\ensuremath{\\mathbb{Z}}^n$,\n$\\rowv{x}^{\\colv{a}} = x_1^{a_1} \\, \\cdots \\, x_n^{a_n}$\nis a \\emph{Laurent monomial} in $\\rowv{x}$\nwith the convention that $x^0 = 1$ (including $x=0$).\nA \\emph{Laurent polynomial}, $f$, is a linear combination of Laurent monomials and\nits \\emph{support} and \\emph{Newton polytope} are denoted\n$\\ensuremath{\\operatorname{supp}}(f)$ and $\\ensuremath{\\operatorname{Newt}}(f)$, respectively.\nWith respect to a vector $\\rowv{v}$,\nthe \\emph{initial form} of $f$ \nis the Laurent polynomial\n$\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(f)(\\rowv{x}) := \\sum_{\\colv{a} \\in (S)_{\\colv{v}}} c_{\\colv{a}} \\, \\rowv{x}^{\\colv{a}}$, \nwhere $(S)_{\\colv{v}}$ is the subset of $S$ on which \n$\\inner{ \\rowv{v} }{ \\cdot }$ is minimized.\nFor an integer matrix $A = [ \\colv{a}_1 \\; \\cdots \\; \\colv{a}_m]$,\n$ \\rowv{x}^A := ( \\rowv{x}^{\\colv{a}_1}, \\dots, \\rowv{x}^{\\colv{a}_m} )$\nis a system of Laurent monomials,\nand we restrict the domain to\nthe \\emph{algebraic torus} $(\\ensuremath{\\mathbb{C}}^*)^n = (\\ensuremath{\\mathbb{C}} \\setminus \\{0\\})^n$\nwhose group structure is given by \n$(x_1,\\dots,x_n) \\circ (y_1,\\dots,y_n) := (x_1 y_1, \\dots, x_n y_n)$.\n\n\nWe fix $G$ to be a connected graph\nwith vertex set $\\ensuremath{\\mathcal{V}}(G) = \\{0,1,\\dots,n\\}$\nand edge set $\\ensuremath{\\mathcal{E}}(G) \\ne \\varnothing$.\nFor node $i$ of $G$, $\\ensuremath{\\mathcal{N}}_G(i)$ is the set of its adjacent nodes.\nArrowheads will be used to distinguish digraphs from graphs,\ne.g., $\\dig{G}$.\nA more thorough list of notation is included in \\Cref{app: notations}.\n\n\n\\subsection{The Kuramoto model}\n\nA network of $n+1$ coupled oscillators can be modeled by a graph $G$ \nwith nodes $\\ensuremath{\\mathcal{V}}(G) = \\{ 0, \\dots, n \\}$ and edges $\\ensuremath{\\mathcal{E}}(G)$ \nrepresenting the oscillators and their connections, respectively.\nOscillators have natural frequencies $\\colv{w} = (w_0,w_1,\\ldots,w_n)^\\top$,\nand, along the edges in $G$, nonzero constants $K = \\{ k_{ij} \\}$\nwith $k_{ij} = k_{ji}$ quantify the coupling strength.\nThe structure $(G,K,\\colv{w})$ encoding this model\nwill simply be called a \\term{network}.\nThe dynamics are described by \n\\begin{equation}\\label{equ: kuramoto ode}\n    \\frac{d \\theta_i}{dt} =\n    w_i - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_{G}(i)} k_{ij} \\sin(\\theta_i - \\theta_j) \n    \\quad\\text{for } i = 0,\\dots,n, \n\\end{equation}\nwhere \n$\\theta_i$ is the phase angle of the $i$-th oscillator \\cite{Kuramoto1975Self}.\n\\emph{Frequency synchronization configurations}\nare defined to be configurations of $(\\theta_0,\\dots,\\theta_n)$ \nfor which all oscillators are tuned to have the exact same angular velocity\n--- the average of the natural frequencies $\\overline{w}$.\nBy adopting a rotational frame,\nwe can assume $\\theta_0 = 0$.\nSince $k_{ij} = k_{ji}$,\nwe can also eliminate one equation.\nTherefore, the (frequency) synchronization configurations\nare the zeroes to the system of $n$ transcendental functions:\n\\begin{equation}\\label{equ: kuramoto sin}\n    (w_i - \\overline{w}) - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_{G}(i)} k_{ij} \\sin(\\theta_i - \\theta_j) \n    \\quad\\text{for } i = 1,\\dots,n.\n\\end{equation}\nThe central question of finding the\nmaximum number of synchronization configurations of a Kuramoto network\nis therefore equivalent to the root counting question for this system.\n\n\\subsection{Algebraic Kuramoto equations}\n\nTo leverage the power of root counting techniques from algebraic geometry,\nthe above transcendental system can be reformulated into an algebraic system\nvia the change of variables $x_i = e^{\\ensuremath{\\mathfrak{i}} \\theta_i}$\nwhere $\\ensuremath{\\mathfrak{i}} = \\sqrt{-1}$.\nThen $\\sin(\\theta_i - \\theta_j) =  \\frac{1}{2 \\ensuremath{\\mathfrak{i}}}(\\frac{x_i}{x_j} - \\frac{x_j}{x_i})$,\nand~\\eqref{equ: kuramoto sin} is transformed into a system of \n$n$ Laurent polynomials $\\colv{f}_G = (f_{G,1},\\dots,f_{G,n})^\\top$, given by \n\\begin{equation}\\label{equ: algebraic kuramoto}\n    f_{G,i}(x_1,\\dots,x_n) = \n    \\overline{w}_i - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_{G}(i)} \n        a_{ij}\n        \\left(\n            \\frac{x_i}{x_j} - \\frac{x_j}{x_i}\n        \\right)\n    \\quad \\text{for } i = 1,\\dots,n,\n\\end{equation}\nwhere $a_{ji} = a_{ij} = \\frac{k_{ij}}{2\\ensuremath{\\mathfrak{i}}}$,\nand $\\overline{w}_i = w_i - \\overline{w}$.\nThis system will be referred to as the\n\\term{algebraic Kuramoto system},\\footnote{%\n    $f_G$ depends on the choice of\n    coupling coefficients $K = [k_{ij}]$ and the natural frequencies $\\colv{w}$,\n    and we will use the notation\n    $f_{(G,K)}$ or $f_{(G,K,\\colv{w})}$\n    to emphasize these dependencies, when needed.\n}\nand it captures all synchronization configurations in the sense that \nthe real zeros to~\\eqref{equ: kuramoto sin} correspond to the\ncomplex zeros of~\\eqref{equ: algebraic kuramoto} \non the real torus $(S^1)^n$\n(i.e., $|x_i| = |e^{\\ensuremath{\\mathfrak{i}} \\theta}| = 1$).\n\nIn much of this paper, we relax the root-counting problem\nby considering all $\\ensuremath{\\mathbb{C}}^*$-zeros of \\eqref{equ: algebraic kuramoto}.\\footnote{\n    Throughout this paper, we talk about $\\ensuremath{\\mathbb{C}}^*$-zeros of \\eqref{equ: algebraic kuramoto},\n    even though there is no difference between $\\ensuremath{\\mathbb{C}}^*$-zeros and $\\ensuremath{\\mathbb{C}}$-zeros.\n    We emphasize this distinction because in recent algebraic studies of\n    Kuramoto equations, other algebraic formulations of \\eqref{equ: kuramoto sin}\n    have been used, and the phrase ``complex zeros\/roots of the Kuramoto equations''\n    has a broader meaning, which include zeros at ``toric infinity'' in\n    the formulation used in this paper.\n}\nOne important observation\nis that we can ignore pendant nodes $G$\nin the sense that synchronization configurations naturally extend to pendant nodes.\n\n\\begin{lemma}\\cite[Theorem 2.5.1]{juliathesis}\\label{lem: leaf extension}\n     Suppose $v \\ne 0$ is a pendant node of $G$.\n    Let $G' = G - \\{ v \\}$.\n    Then any $\\ensuremath{\\mathbb{C}}^*$-zero of $\\colv{f}_{G'}$\n    extends to two distinct $\\ensuremath{\\mathbb{C}}^*$-zeros for $\\colv{f}_G$.\n\\end{lemma}\n\n\n\\subsection{Power flow equations}\\label{sec: power flow}\n\nThe \\emph{PV-type power flow system} is one important variation\nof the Kuramoto system.\nIn it, the graph $G$ models an electric power network\nwhere $\\ensuremath{\\mathcal{V}}(G) = \\{ 0, \\ldots, n \\}$ represent\nbuses in the power network.\nAn edge $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$, representing the connection between buses $i$ and $j$,\nhas a known complex admittance $b_{ij}' + \\ensuremath{\\mathfrak{i}} g_{ij}'$.\nFor each bus $i$, the relationship between its complex power injection $P_i + \\ensuremath{\\mathfrak{i}} Q_i$\nand the complex voltages is\ncaptured by the nonlinear equations\n\\begin{align}\n    P_i &= \\sum_{j \\in \\mathcal{N}_G(i)} |V_i||V_j| (g_{ij}' \\cos(\\theta_i - \\theta_j) + b_{ij}' \\sin (\\theta_i - \\theta_j) ) \\label{trigpfeqs1} \\\\\n    Q_i &= \\sum_{j \\in \\mathcal{N}_G(i)} |V_i||V_j| (g_{ij}' \\sin (\\theta_i - \\theta_j) - b_{ij}' \\cos(\\theta_i - \\theta_j)) \\label{trigpfeqs2}\n\\end{align}\nwhere $V_i$ and $\\theta_i$ are the magnitude and the phase angle\nof the complex voltage at bus $i$.\nWe fix bus $0$ to be the \\emph{slack bus}, meaning $\\theta_0 = 0$.\nEquations \\eqref{trigpfeqs1}-\\eqref{trigpfeqs2} are the \\emph{power flow equations}.  \n\nThe case where\n$Q_i$ and $\\theta_i$ are unknown while $P_i$ and $|V_i|$ are known constants\nare called PV nodes and they model typical generator buses.\nAs above, through the change of variables $x_i = e^{\\ensuremath{\\mathfrak{i}} \\theta_i}$\nwe get the \\emph{PV-type algebraic power flow equations}\n\\begin{align}\n   f_{G,i}(x_1,\\ldots,x_n)&=  P_i -  \\sum_{j \\in \\mathcal{N}_G(i)} g_{ij} \\left(\\frac{x_i}{x_j} + \\frac{x_j}{x_i}\\right)  + b_{ij}\\left(\\frac{x_i}{x_j} - \\frac{x_j}{x_i}\\right), \\quad \\text{for } i = 1,\\ldots,n \\label{eq:pfeqs}\n\\end{align}\nwhere $b_{ij} = \\frac{1}{2\\ensuremath{\\mathfrak{i}}}|V_i||V_j|b_{ij}'$ and $g_{ij} = \\frac{1}{2}|V_i||V_j|g_{ij}'$.\nWhen $g_{ij} = 0$, the corresponding power system is \\emph{lossless}\nand \\eqref{eq:pfeqs} reduces to \\eqref{equ: algebraic kuramoto}.\nOtherwise, the system is \\emph{lossy}\nand \\eqref{eq:pfeqs} differs from \\eqref{equ: algebraic kuramoto}\nin the constraints on the coefficients.\nYet, the same set of monomials are involved,\nand as we will demonstrate,\nthe algebraic arguments we will develop\ncan be applied to this generalization.\n\n\n\n\\subsection{Kuramoto equations with phase delays}\\label{sec: phase delays}\n\nAnother generalization of the Kuramoto model considers \\emph{phase delays}.\nWith known phase delay parameters $\\delta_{ij} \\in \\mathbb{R}$,\n \\eqref{equ: kuramoto sin} is augmented to:\n\\[\n    0 = w_i - \\overline{w} -\n    \\sum_{j \\in \\mathcal{N}_G(i)} k_{ij}\n    \\sin(\\theta_i - \\theta_j + \\delta_{ij}), \\quad \\text{for } i = 1,\\ldots,n,\n\\]\nwhich corresponds to the system in which for a pair of coupled oscillators $i$ and $j$,\noscillator $i$ responds not directly to the phase angle of oscillator $j$\nbut its delayed phase $\\theta_j - \\delta_{ij}$.\nLetting $C_{ij} = e^{{\\mathbf{i}} \\delta_{ij}}$, we can again make this system algebraic giving:\n\\begin{align}\\label{equ: delayed Kuramoto}\n    f_{G,i}(x_1,\\ldots,x_n) =\n    \\overline{w}_i - \\sum_{j \\in \\mathcal{N}_G(i)} a_{ij} \\left( \\frac{x_i C_{ij}}{x_j} - \\frac{x_j}{x_i C_{ij}} \\right), \\quad \\text{for } i = 1,\\ldots,n.\n\\end{align}\nThis is, again, a generalization of the algebraic Kuramoto system \\eqref{equ: algebraic kuramoto}\nthat involves the same set of monomials.\nThe algebraic results to be developed also apply to this family of algebraic systems.\n\n\n\\subsection{BKK bound}\n\nThe root counting arguments in this paper revolve around the\nBernshtein-Kushnirenko-Khovanskii (BKK) bound,\nespecially Bernshtein's Second Theorem.\n\n\\begin{theorem}[D. Bernshtein 1975~\\cite{Bernshtein1975Number}]\\label{thm:bernshtein-b}\n    (A) For a square Laurent system $\\colv{f} = (f_1,\\dots,f_n)$,\n    if for all nonzero vectors $\\colv{v} \\in \\ensuremath{\\mathbb{R}}^n$,\n    $\\ensuremath{\\operatorname{init}}_{\\colv{v}}(\\colv{f})$ has no $\\ensuremath{\\mathbb{C}}^*$-zeros,\n    then all $\\ensuremath{\\mathbb{C}}^*$-zeros of $\\colv{f}$ are isolated,\n    and the total number, counting multiplicity,\n    is the mixed volume $M = \\operatorname{MV}(\\ensuremath{\\operatorname{Newt}}(f_1),\\ldots,\\ensuremath{\\operatorname{Newt}}(f_n))$.\n    \n    (B) If $\\ensuremath{\\operatorname{init}}_{\\colv{v}} (\\colv{f})$ has a $\\ensuremath{\\mathbb{C}}^*$-zero\n    for some $\\colv{v} \\ne \\colv{0}$,\n    then the number of isolated $\\ensuremath{\\mathbb{C}}^*$-zeros $\\colv{f}$ has,\n    counting multiplicity, is strictly less than $M$ if $M>0$.\n\\end{theorem}\nA system for which the condition (A) holds\nis said to be \\emph{Bernshtein-general}.\nOnly the special case of identical Newton polytopes,\ni.e., when $\\ensuremath{\\operatorname{Newt}}(f_1),\\ldots,\\ensuremath{\\operatorname{Newt}}(f_n)$ are all identical,\nwill be used.\nThis specialized version strengthens\nKushnirenko's Theorem \\cite{Kushnirenko1975Newton}.\n\n\n\n\\subsection{Randomized algebraic Kuramoto system}\\label{sec: randomized}\n\nThe analysis of the algebraic Kuramoto system~\\eqref{equ: algebraic kuramoto}\ncan be further simplified through ``randomization''\n(a.k.a. ``pre-conditioning'').\nTaking $\\colv{f}_G$ as a column vector,\nfor any nonsingular matrix $R$,\nthe systems $\\colv{f}_G$ and $\\colv{f}^*_G := R \\, \\colv{f}_G$ have the same zero set.\nWith a generic choice of $R$, there will be no complete cancellation of terms, \nand $\\colv{f}^*_G$\nwill be referred to as the \\term{randomized (algebraic) Kuramoto system}.\nThis system is \\emph{unmixed} in the sense that\nfor all $i \\in [n]$, $f^*_{G,i}$ have identical supports\nsince they involve the same set of monomials, namely, constant terms, $x_i x_j^{-1}$ and $x_j x_i^{-1}$ for every edge $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$.\n\n\nThe randomization $\\colv{f}_G \\mapsto \\colv{f}^*_G$ does not alter the zero set\nbut makes a very helpful change to the tropical structure in the sense\nthat much is already known about\nthe tropical self-intersection of $\\colv{f}^*_G$.\nIn particular, there is a mapping between the\ngraph-theoretical features of $G$\nand the tropical structures of $\\colv{f}^*_G$\nthrough which we can gain key insight into the structure\nof the zeros of $\\colv{f}^*_G$.\nThe next three subsections briefly review the geometric interpretation of this connection.\n\n\n\\subsection{Adjacency polytopes}\n\nFor each $i=1,\\ldots,n$, $\\ensuremath{\\operatorname{supp}}(f^*_{G,i})$ are all identical\nand given by\n\\[\n    \\ensuremath{\\check{\\nabla}}_G :=\n    \\{ \n        \\pm (\\colv{e}_i - \\colv{e}_j) \\mid \n        \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)\n    \\}\n    \\;\\cup\\;\n    \\{ \\colv{0} \\},\n\\]\nwhere $\\colv{e}_i$ is the $i$-th standard basis vector of $\\ensuremath{\\mathbb{R}}^n$ for $i=1,\\ldots,n$,\nand $\\colv{e}_0 = \\colv{0}$.\nIn Refs.~\\cite{Chen2019Unmixing,ChenDavisMehta2018Counting},\nthe convex hull of $\\ensuremath{\\check{\\nabla}}_G$ is called the \\emph{adjacency polytope} of $G$\n(of PV-type\\footnote{%\n    This is retroactively named the\n    adjacency polytope of PV-type to distinguish it from\n    those of PQ-type \\cite{ChenMehta2017Network}.\n    In general, an adjacency polytope associated with a graph $G$\n    is the convex hull of $\\{ \\colv{0} \\}$\n    and point sets $\\{ P_{ij} \\mid \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G) \\}$\n    that depend on the ``types'' of the nodes,\n    a concept originally derived from the power-flow studies\n    (\\Cref{sec: power flow}).\n}).\nIt is unimodularly equivalent to the \\emph{symmetric edge polytope} of $G$\nthat has appeared in number theory and discrete geometry\n(see, for example, the broad overview provided in \\cite{DAliDelucchiMichalek2022Many}).\n\n\nWe will not distinguish $\\ensuremath{\\check{\\nabla}}_G$ from its convex hull, meaning a ``face'' of $\\ensuremath{\\check{\\nabla}}_G$ refers to a subset $F \\subseteq \\ensuremath{\\check{\\nabla}}_G$\nsuch that $\\conv(F)$ is a face of $\\conv(\\ensuremath{\\check{\\nabla}}_G)$\nand $\\dim(F) := \\dim(\\conv(F))$.\nThe set of facets and the boundary of $\\ensuremath{\\check{\\nabla}}_G$ are denoted\n$\\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G)$ and $\\partial \\ensuremath{\\check{\\nabla}}_G$, respectively.\nThe \\emph{corank}\\footnote{%\n    In matroid theory, the more common term is ``\\emph{nullity}''.\n    Here, we follow the convention from convex geometry.\n}\nof a face $F$, denoted $\\operatorname{corank}(F)$, is the number $|F| - \\dim(F) - 1$.\n\nBy Kushnirenko's Theorem \\cite{Kushnirenko1975Newton},\nthe \\emph{normalized volume} \nof the adjacency polytope, denoted $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$,\nis an upper bound for the $\\ensuremath{\\mathbb{C}}^*$-zero count\nfor $\\colv{f}^*_G$ and $\\colv{f}_G$\nwhich also bounds the real zero count to the\ntranscendental Kuramoto system \\eqref{equ: kuramoto sin}.\nThis is the \\emph{adjacency polytope bound}\n\\cite{Chen2019Unmixing,ChenDavisMehta2018Counting}.\n\n\n\\subsection{Faces and face subgraphs}\n\n\nThere is an intimate connection between faces of $\\ensuremath{\\check{\\nabla}}_G$\nand subgraphs of $G$.\nSince $\\colv{0}$ is an interior point of $\\ensuremath{\\check{\\nabla}}_G$,\nevery vertex of a proper face $F$\nof $\\ensuremath{\\check{\\nabla}}_G$\nis of the form $\\colv{e}_i - \\colv{e}_j$ for some $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$.\nThus, it is natural to consider the corresponding\n\\emph{facial subgraph} $G_F$ and \n\\emph{facial subdigraph} $\\dig{G}_F$, given by\n\\begin{align*}\n    \\ensuremath{\\mathcal{E}}(G_F)       &= \\{ \\{i,j\\} \\mid \\colv{e}_i - \\colv{e}_j \\in F \\text{ or } \\colv{e}_j - \\colv{e}_i \\in F \\} \n    \\;\\text{and} \\\\\n    \\ensuremath{\\mathcal{E}}(\\dig{G}_F) &= \\{  (i,j)  \\mid \\colv{e}_i - \\colv{e}_j \\in F \\}.\n\\end{align*}\nAs defined in \\cite{Chen2019Directed,DAliDelucchiMichalek2022Many}, for $F \\in \\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G)$,\n$\\ensuremath{\\mathcal{E}}(G_F)$ is called a \\emph{facet subgraph} and $\\ensuremath{\\mathcal{E}}(\\dig{G}_F)$ a \\emph{facet subdigraph}.\nHigashitani, Jochemko, and Micha\\l{}ek\nprovided a topological classification of face subgraphs\n\\cite[Theorem 3.1]{HigashitaniJochemkoMichalek2019Arithmetic}\nand it was later reinterpreted \\cite[Theorem 3]{ChenDavisKorchevskaia2022Facets}.\nWe state the latter here.\n\n\\begin{theorem}[Theorem 3 \\cite{ChenDavisKorchevskaia2022Facets}]\\label{thm: faces are max bipartite}\n    Let $H$ be a nontrivial connected subgraph of $G$.\n    \\begin{enumerate}\n        \\item $H$ is a face  subgraph of $G$ if and only if it is a maximal bipartite subgraph of $G[\\ensuremath{\\mathcal{V}}(H)]$.\n        \\item $H$ is a facet subgraph of $G$ if and only if it is a maximal bipartite subgraph of $G$.\n    \\end{enumerate}\n\\end{theorem}\n\nMultiple faces can correspond to the same facial subgraph\n(in particular, $G_F = G_{(-F)}$ for any face $F$).\nThe crisper parametrization is given by the correspondence\n$F \\mapsto \\dig{G}_F$.\nRef.~\\cite{ChenDavisKorchevskaia2022Facets} describes\nthe balancing conditions that characterize facial subdigraphs.\nHere, we only make use of the balancing condition\nfor the ``primitive cycle vectors'' associated with a face subdigraph.\n\n\nFor a facial subdigraph $\\dig{G}_F$, its \\emph{reduced incidence matrix}\n$\\ensuremath{\\check{Q}}(\\dig{G}_F)$ is the matrix with columns $\\colv{e}_i - \\colv{e}_j$\nsuch that $(i,j) \\in \\ensuremath{\\mathcal{E}}(\\dig{G}_F)$\n(see \\Cref{app: notations}).\nFor a face $F$ of $\\ensuremath{\\check{\\nabla}}_G$ such that $G_F$ is connected,\nits corank $d$ equals the nullity of $\\ensuremath{\\check{Q}}(\\dig{G}_F)$.\nThe null space is spanned by $d$ vectors with entries $\\{ +1, 0, -1 \\}$,\nindicating the incidence of arcs with the cycles in $G_F$ with prescribed orientations.\nThese vectors will be referred to as \\emph{primitive cycle vectors}.\n\n\n\\begin{lemma}[Theorem 7 \\cite{ChenDavisKorchevskaia2022Facets} ]\\label{lem: primitive null vector}\n    Let $F$ be a face of $\\ensuremath{\\check{\\nabla}}_G$ for which $G_F$ is connected.\n    For any cycle in $G_F$,\n    let $\\colv{\\eta}$\n    be its associated primitive cycle vector with respect to $\\dig{G}_F$,\n    then\n        $\\innerv{ 1 }{ \\eta } = 0$.\n\\end{lemma}\n\nNote that these conditions are only necessary conditions\non the primitive cycle vectors.\nIf $G$ contains more than one cycle, these conditions are not sufficient.\nStronger classifications results (which are not needed here)\ncan be found in \\cite{ChenDavisKorchevskaia2022Facets,HigashitaniJochemkoMichalek2019Arithmetic}.\n\nFor a subdigraph $\\dig{H}$ of $\\dig{G}$,\nthe \\emph{coupling vector} $\\rowv{k}(\\dig{H})$ has entries \n$k_{ij}$ for $(i,j) \\in \\dig{H}$.\nSimilarly, the entries of $\\rowv{a}(\\dig{H})$ are\n$a_{ij} = \\frac{k_{ij}}{2 \\ensuremath{\\mathfrak{i}}}$ \nfor $(i,j) \\in \\ensuremath{\\mathcal{E}}(\\dig{H})$.\nThe ordering of the entries is arbitrary,\nbut when appearing in the same context with $\\ensuremath{\\check{Q}}(\\dig{H})$,\nconsistent ordering is implied.\n\n\n\\subsection{Facial systems}\n\nThe vast literature on the facial structure of $\\ensuremath{\\check{\\nabla}}_G$\ngives us a shortcut to understanding the initial systems\nof the randomized algebraic Kuramoto system $\\colv{f}^*_G$.\nRecall that $\\ensuremath{\\operatorname{supp}}(f^*_{G,i})$ are all identical,\nso the initial systems of $\\colv{f}^*_G$ have particularly simple descriptions\ncorresponding to proper faces of $\\ensuremath{\\check{\\nabla}}_G$.\nFor any $0 \\neq \\rowv{v} \\in \\ensuremath{\\mathbb{R}}^n$,\nthe initial system $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(\\colv{f}^*_G)$ is\n\\begin{equation}\\label{equ: facial system}\n    \\ensuremath{\\operatorname{init}}_{\\rowv{v}}(f^*_{G,i})(\\rowv{x}) =\n    \\sum_{ \\colv{e}_j - \\colv{e}_{j'} \\in F } c_{i,j,j'} \\, \\rowv{x}^{\\colv{e}_j - \\colv{e}_{j'}} =\n    \\sum_{ (j,j') \\in \\ensuremath{\\mathcal{E}}(\\dig{G}_F) } c_{i,j,j'} \\, \\rowv{x}^{\\colv{e}_j - \\colv{e}_{j'}}\n    \\quad\\text{for } i = 1,\\ldots,n,\n\\end{equation}\nwhere $F$ is the proper face of $\\ensuremath{\\check{\\nabla}}_G$ for which\n$\\rowv{v}$ is an inner normal vector.\nWe will make frequent use of this geometric interpretation and\ntherefore it is convenient to slightly abuse the notation and write\n$\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G) := \\ensuremath{\\operatorname{init}}_{\\colv{v}} (\\colv{f}^*_G)$\nwhen the particular choice of vector $\\colv{v}$ defining the face $F$ is not important.\nIt will be called a \\emph{facial system} of $\\colv{f}^*_G$,\n(or a \\emph{facet system} if $F \\in \\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G)$).\n\n\\section{Generic complex root count}\\label{sec: generic root count}\n\n\nIn this section, we establish the $\\ensuremath{\\mathbb{C}}^*$-zero count for the algebraic Kuramoto system \\eqref{equ: algebraic kuramoto}\nfor generic choices of real constants $\\overline{w}_1,\\ldots,\\overline{w}_n$\nand $\\{ k_{ij} \\}_{\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)}$.\nWe show that the generic $\\ensuremath{\\mathbb{C}}^*$-zero count is the adjacency polytope bound\n$\\nvol(\\ensuremath{\\check{\\nabla}}_G)$. \nA corollary is that this system is\nBernshtein-general, despite the constraints on the coefficients.\n\nThe first observation is that since $\\ensuremath{\\mathbb{R}}$ is Zariski-dense in $\\ensuremath{\\mathbb{C}}$, \ngeneric choices of complex parameters can be used without changing the $\\ensuremath{\\mathbb{C}}^*$-zero count.\nThat is, it is sufficient to focus on the root counting problem for the system\n\\[\n    0 = \\overline{w}_i - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_{G}(i)} \n        a_{ij}\n        \\left(\n            \\frac{x_i}{x_j} - \\frac{x_j}{x_i}\n        \\right)\n    \\qquad \\text{for } i = 1,\\dots,n,\n\\]\nin the $n$ unknowns $x_1,\\ldots,x_n$ (with $x_0=1$)\nfor generic \\emph{complex} choices\\footnote{%\n    For simplicity, we will assume the constant terms\n    $\\overline{w}_1,\\ldots,\\overline{w}_n$ are chosen generically.\n    This is equivalent to choosing $w_1,\\ldots,w_n$ generically\n    and leaving $w_0$ constrained by these choices.} of $\\overline{w}_i,a_{ij}$.\n\nThere are three main obstacles.\nFirst, in each polynomial the monomials $x_i\/x_j$ and $x_j\/x_i$ share the same coefficient $a_{ij}$.\nSecond, for any edge $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$,\nthe $i$-th and $j$-th polynomials have the terms\n$a_{ij} (x_i x_j^{-1} - x_j x_i^{-1})$\nand\n$a_{ji} (x_j x_i^{-1} - x_i x_j^{-1})$,\nrespectively, which are negations of each other\nsince $a_{ij} = a_{ji}$.\nTherefore, the allowed choices of coefficients consists of a\nnowhere dense subset of (Lebesgue) measure 0\nin the space of all possible complex coefficients.\n    Finally, unless $G$ is the complete graph,\n    the Newton polytopes of the algebraic Kuramoto system\n    are not full-dimensional\n    which prevents simpler arguments (e.g. \\cite{Chen2018Equality}) from being applied.\nWe will show that despite this,\nthe maximum $\\ensuremath{\\mathbb{C}}^*$-zero count, given by the adjacency polytope bound is generically attained.\n\nBefore presenting the main theorem of this section,\nwe first establish a few technical results that will be used,\nsome of which are well known, but are nonetheless included for completeness.\n\n\n\\begin{lemma}\\label{lem: facet subdivision}\n    The lifting function $\\tilde{\\omega} : \\ensuremath{\\check{\\nabla}}_G \\to \\mathbb{Q}$, given by\n    \\[\n        \\tilde{\\omega}(\\colv{x}) =\n        \\begin{cases}\n            0 & \\text{if } \\colv{x} = \\colv{0} \\\\\n            1 & \\text{otherwise}\n        \\end{cases}\n    \\]\n    induces a regular subdivision\n    $\\Sigma_\\omega(\\ensuremath{\\check{\\nabla}}_G) =  \\{ \\colv{0} \\cup F \\mid F \\in \\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G) \\}$.\n\\end{lemma}\nThis is a well known consequence of assigning a sufficiently small lifting value\nto an interior point ($\\colv{0}$ in this case).\nThe resulting regular subdivision may be too coarse to be useful in our discussions.\nIndeed, it will not be a triangulation unless $G$ is has no even cycles\n\\cite{ChenDavisKorchevskaia2022Facets,ChenDavis2022Toric}.\nIt can be refined into a triangulation through perturbations on\nthe nonzero lifting values.\n\n\\begin{lemma}\\label{lem: generic lifting}\n    For generic but symmetric choices\n    $\\{ \\delta_{ij} = \\delta_{ji} \\in \\ensuremath{\\mathbb{R}} \\mid \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G) \\}$\n    that are sufficiently close to 0, the function\n    $\\omega : \\ensuremath{\\check{\\nabla}}_G \\to \\mathbb{Q}$, given by\n    \\[\n        \\omega(\\colv{a}) =\n        \\begin{cases}\n            0               & \\text{if } \\colv{a} = \\colv{0} \\\\\n            1 + \\delta_{ij} & \\text{if } \\colv{a} = \\colv{e}_i - \\colv{e}_j\n        \\end{cases}\n    \\]\n    induces a regular unimodular triangulation\n    $\\Delta_\\omega = \\Sigma_\\omega(\\ensuremath{\\check{\\nabla}}_G)$\n    that is a refinement of $\\Sigma_{\\tilde{w}}(\\ensuremath{\\check{\\nabla}}_G)$\n    and its cells are in one-to-one correspondence with cells in\n    a unimodular triangulation of \\(\\partial \\ensuremath{\\check{\\nabla}}_G\\).\n    Indeed, each cell is of the form\n    $0 \\cup \\Delta$ where $\\Delta$ is a simplex in $\\partial \\ensuremath{\\check{\\nabla}}_G$.\n\\end{lemma}\n\n\\begin{proof}\n    We first show the interior point $\\colv{0}$ is contained in every cell. \n    Let $C \\in \\Delta_\\omega$\n    and $(\\colv{v},1)$ be the inner normal vector the lower facet of $\\ensuremath{\\check{\\nabla}}_G^\\omega$\n    whose projection is $C$.\n    Suppose $\\colv{0} \\not\\in C$, \n    then $C$ contains an affinely independent set of $n+1$ points\n    $\\{ \\colv{a}_0,\\ldots,\\colv{a}_n \\} \\not\\ni \\colv{0}$,\n    and $\\colv{v}$ satisfies the equation\n    \\[\n        \\begin{bmatrix}\n            \\colv{a}_1^\\top - \\colv{a}_0^\\top \\\\\n            \\vdots \\\\\n            \\colv{a}_n^\\top - \\colv{a}_0^\\top \\\\\n        \\end{bmatrix}\n        \\,\n        \\colv{v}\n        =\n        \\begin{bmatrix}\n            \\omega(\\colv{a}_0) - \\omega(\\colv{a}_1) \\\\\n            \\vdots \\\\\n            \\omega(\\colv{a}_0) - \\omega(\\colv{a}_n)\n        \\end{bmatrix}.\n    \\]\n    Let $B$ be the matrix on the left and $\\colv{\\beta}$ be the vector on the right,\n    then $\\colv{v} = B^{-1} \\colv{\\beta}$ and thus\n    \\[\n        | \\inner{ \\colv{a} }{ \\colv{v} } | \\le \n        \\| \\colv{a} \\| \\| \\colv{v} \\| \\le\n        \\| \\colv{a} \\| \\, \\| B^{-1} \\| \\, \\| \\colv{\\beta} \\|\n        \\quad\\text{for any } \\colv{a} \\in C.\n    \\]\n    Note that entries of $\\colv{\\beta}$ are differences among the\n    $\\{ \\delta_{ij} \\}$.\n    Therefore, for any $\\epsilon > 0$, there is a $\\delta$ such that\n    $\\delta_{ij} < \\delta$ for all $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$\n    implies $| \\inner{ \\colv{a} }{ \\colv{v} } | < \\epsilon$,\n    contradicting with the assumption that\n    \\[\n        \\inner{ \\colv{v} }{ \\colv{a} } + \\omega(\\colv{a}) <\n        \\sinner{ \\colv{v} }{ \\colv{0} } + \\omega(\\colv{0}) = 0.\n    \\]\n    This shows that $\\colv{0}$ must be contained in every cell.\n    \n\n    To establish $\\Delta_\\omega$ as a triangulation,\n    it is sufficient to show nonzero points in a cell $C \\in \\Delta_\\omega$\n    are assigned independent lifting values by $\\omega$.\n    If $\\pm (\\colv{e}_i - \\colv{e}_j) \\in C$, for some $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$,\n    then\n    \\[\n        \\inner{ \\colv{v} }{ \\pm (\\colv{e}_i - \\colv{e}_j) } + \n        \\omega( \\pm (\\colv{e}_i - \\colv{e}_j) ) = h,\n    \\]\n    where $h$ is the minimum of $\\inner{ \\colv{v} }{ \\,\\cdot\\, } + \\omega(\\,\\cdot\\,)$\n    over $\\ensuremath{\\check{\\nabla}}_G$.\n    Since $\\omega(\\pm (\\colv{e}_i - \\colv{e}_j)) = 1 + \\delta_{ij}$,\n    summing the two equation produces\n    \\[\n        1 + \\delta_{ij} = h > 0,\n    \\]\n    which contradict with constraint that\n    $0 = \\sinner{\\colv{v}}{\\colv{0}} + \\omega(\\colv{0}) \\ge h$.\n    Therefore, if $\\colv{e}_i - \\colv{e}_j \\in C$,\n    then $\\colv{e}_j - \\colv{e}_i \\not\\in C$.\n    Consequently, the nonzero points in $C$ are associated with\n    independent generic choices of lifting values,\n    and thus $C$ must be a simplex.\n    \n\n    For unimodularity, note that, since\n    $\\colv{e}_i - \\colv{e}_j \\in C$ implies $\\colv{e}_j - \\colv{e}_i \\not\\in C$\n    for $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$ and for a cell $C \\in \\Delta_\\omega$,\n    the nonzero points of $C$, as vectors, are exactly columns of (signed) incidence matrix of $G$,\n    which is unimodular.\n    Therefore, each cell of $\\Delta_\\omega$ is unimodular.\n    \n\n    Finally, we show that $\\Delta_\\omega$ is a refinement of\n    $\\Sigma_{\\tilde{\\omega}}(\\ensuremath{\\check{\\nabla}})$ from \\Cref{lem: facet subdivision}.\n    In the following, we shall fix an arbitrary ordering of the points in $\\ensuremath{\\check{\\nabla}}_G$\n    and consider lifting functions on $\\ensuremath{\\check{\\nabla}}_G$,\n    e.g., $\\omega$ and $\\tilde{\\omega}$,\n    as vectors in $\\ensuremath{\\mathbb{R}}^{|\\ensuremath{\\check{\\nabla}}_G|}$, whose entries are their lifting values.\n    Let $\\mathcal{C} = \\mathcal{C}(\\ensuremath{\\check{\\nabla}}_G,\\Delta_\\omega)$ be the (closed) secondary cone\n    of $\\Delta_\\omega$ in $\\ensuremath{\\check{\\nabla}}_G$.\n    Then $\\mathcal{C}$ is full-dimensional,\n    since $\\Delta_\\omega$ is a triangulation.\n    By assumption, $\\omega$ is sufficiently close to $\\tilde{\\omega}$,\n    so $\\tilde{\\omega} \\in \\mathcal{C}$.\n    Consequently, $\\Delta_\\omega = \\Sigma_\\omega(\\ensuremath{\\check{\\nabla}}_G)$\n    equals or refines $\\Sigma_{\\tilde{\\omega}}(\\ensuremath{\\check{\\nabla}}_G)$.\n\\end{proof}\n\nA lifting function \\(\\omega : \\ensuremath{\\check{\\nabla}}_G \\to \\mathbb{Q}\\) for which \\Cref{lem: generic lifting}\nholds will be referred to as a \\term{generic symmetric lifting function} for $\\ensuremath{\\check{\\nabla}}_G$.\n\n\\begin{restatable}{lemma}{simplextree}\\label{lem: simplex tree}\n    For a generic symmetric lifting function $\\omega$ for $\\ensuremath{\\check{\\nabla}}_G$,\n    let $\\Delta$ be a simplex in $\\Delta_\\omega$.\n    Then the digraph $\\dig{G}_\\Delta$\n    is acyclic, and its underlying graph $G_\\Delta$\n    is a spanning tree of $G$.\n\\end{restatable}\n\nThe proof of this lemma is nearly identical to the proof of\n\\cite[Theorem 1]{Chen2019Directed}\nand we include an elementary proof in \\Cref{sec: lemmas} for completeness.\n\n\\begin{lemma}\\label{lem: tree system}\n    Suppose $\\dig{T} < \\dig{G}$ is a acyclic,\n    and its underlying graph $T$ is a spanning tree of $G$,\n    then, for generic $\\overline{w}_1,\\ldots,\\overline{w}_n \\in \\ensuremath{\\mathbb{C}}^*$\n    and \\emph{any} choices of $a_{ij} \\in \\ensuremath{\\mathbb{C}}^*$,\n    the system of $n$ Laurent polynomials\n    \\[\n        \\overline{w}_i\n        - \\sum_{j \\in \\mathcal{N}^+_{\\vec{T}}(i)} a_{ij} \\frac{x_i}{x_j}\n        + \\sum_{j \\in \\mathcal{N}^-_{\\vec{T}}(i)} a_{ij} \\frac{x_j}{x_i}\n        \\quad\\text{for } i = 1,\\ldots,n\n    \\]\n    has a unique zero in $(\\ensuremath{\\mathbb{C}}^*)^n$,\n    and this zero is isolated and regular.\n\\end{lemma}\n\n\n\nHere,\n\\(\\mathcal{N}^+_{\\vec{T}}(i) = \\{ j \\in \\mathcal{V}(\\vec{T}) \\mid (i,j) \\in \\mathcal{E}(\\vec{T}) \\}\\) and\n\\(\\mathcal{N}^-_{\\vec{T}}(i) = \\{ j \\in \\mathcal{V}(\\vec{T}) \\mid (j,i) \\in \\mathcal{E}(\\vec{T}) \\}\\)\nare the sets of adjacent nodes of \\(i\\) through outgoing and incoming arcs, respectively.\n\n\\begin{proof}\n    We prove this by induction on the number of vertices of $G$, $N$. Denote the above system as $\\colv{f}_{\\dig T}(x_0,\\ldots,x_{N-1})$.\n    The statement is  true for the case\n    where $N=2$ and $| \\ensuremath{\\mathcal{V}}(G) | = | \\ensuremath{\\mathcal{V}}(T) | =2$.\n    \n     Assume the statement is true for any graph with $N$ nodes\n    and consider the case $G$ has $N+1$ nodes $\\{0,1,\\ldots,N\\}$.\n    By re-labeling, we can assume node $N$ is a leaf in $T$\n    and is adjacent to node $N-1$.\n    Moreover, since this system is homogeneous of degree 0,\n    we can also scale the homogeneous coordinates\n    so that $x_{N-1} = 1$. \n    Then the above system is decomposed Laurent system\n    $\\colv{f}_{\\dig T'}(x_0,\\ldots,x_{N-1})$\n    and the binomial\n    $\\overline{w}_{N} \\pm a_{N,N-1} x_N^{\\mp 1}$,\n   where $\\dig{T'}$ is the tree $\\dig{T} - \\{ N \\}$.\n   The induction hypothesis, that\n   $\\dig{f}_{\\dig{T'}}(x_0,\\ldots,x_N)$ has a unique isolated and regular\n   thus completes the proof.\n\\end{proof}\n\nWith these technical preparations,\nwe now establish the main theorem of this section.\n\n\\begin{theorem}\\label{thm: generic root count}\n    For generic choices of real or complex constants\n    $\\overline{w}_1,\\ldots,\\overline{w}_n$\n    and generic but symmetric choices of real or complex coupling coefficients\n    $k_{ij} = k_{ji} \\ne 0$ for $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$,\n    the $\\ensuremath{\\mathbb{C}}^*$-zero set of the algebraic Kuramoto system \\eqref{equ: algebraic kuramoto}\n    $\\colv{f}_G = (f_{G,1},\\ldots,f_{G,n})$, given by\n    \\[\n        f_{G,i} (x_1,\\ldots,x_n) =\n        \\overline{w}_i -\n        \\sum_{j \\in \\ensuremath{\\mathcal{N}}_G(i)}\n            \\frac{k_{ij}}{2\\ensuremath{\\mathfrak{i}}}\n            \\left(\n                \\frac{x_i}{x_j} - \\frac{x_j}{x_i}\n            \\right) = 0\n    \\]\n    consists of isolated regular points, and the total number is\n    $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$.\n\\end{theorem}\n\nThis is a far generalization of \n\\cite[Corollary 9 and Theorem 16]{ChenDavisMehta2018Counting} and \n\\cite[Theorem 3.3.3]{GuoSalam1990Determining}\nthe generic $\\ensuremath{\\mathbb{C}}^*$-zero count is established for tree\\footnote{%\n   The generic number of complex equilibria for the Kuramoto model on tree networks is $2^n$.\n    This fact appears to be well known among researchers in power systems\n    long before the referenced paper \\cite{ChenDavisMehta2018Counting}.\n}\ncycle, and complete networks only.\n\n\n\\begin{proof}\n    By Sard's Theorem, for generic choices\n    of $\\overline{w}_1,\\ldots,\\overline{w}_n$, the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_G$\n    consists of isolated and regular points.\n    Under this assumption, we only need to establish the $\\ensuremath{\\mathbb{C}}^*$-zero count.\n    Moreover, since it is already known this zero count is bounded by $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$,\n    it is sufficient to show that it is greater than or equal to this bound.\n    We shall take a constructive approach\n    through a specialized version of the polyhedral homotopy of Huber and Sturmfels\n    \\cite{HuberSturmfels1995Polyhedral}.\n\n    Let $\\omega : \\ensuremath{\\check{\\nabla}}_G \\to \\mathbb{Q}$ be a generic symmetric lifting function\n    for $\\ensuremath{\\check{\\nabla}}_G$. \n    We use the notation $\\omega_{ij} = \\omega(\\colv{e}_i - \\colv{e}_j) = \\omega(\\colv{e}_j - \\colv{e}_i)$\n    and define the function $\\colv{h} = (h_1,\\ldots,h_n) : (\\ensuremath{\\mathbb{C}}^*)^n \\times \\ensuremath{\\mathbb{C}} \\to \\ensuremath{\\mathbb{C}}^n$,\n    given by\n    \\[\n        h_i (x_1,\\ldots,x_n,t) =\n        \\overline{w}_i - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_G(i)}\n        a_{ij} t^{\\omega_{ij}}\n        \\left(\n            \\frac{x_i}{x_j} -\n            \\frac{x_j}{x_i}\n        \\right).\n    \\]\n    Away from $t=0$, $\\colv{h}$ is a parametrized version of the original\n    system $\\colv{f}_G$ with coefficients being\n    analytic functions of the parameter $t$.\n    Note that since $\\omega_{ij} = \\omega_{ji}$, for any choice of\n    $t \\in \\ensuremath{\\mathbb{C}}$, the system still satisfies the symmetry constraints on the coefficients.\n    By the Parameter Homotopy Theorem \\cite{MorganSommese1989Coefficient},\n    for $t$ outside a proper Zariski closed (i.e., finite) set $Q \\subset \\ensuremath{\\mathbb{C}}$,\n    the number of isolated $\\ensuremath{\\mathbb{C}}^*$-zeros of  $\\colv{h}(\\,\\cdot\\,,t) =0$ is a constant\n    which is also an upper bound for the isolated $\\ensuremath{\\mathbb{C}}^*$-zero count\n    for $\\colv{h}(\\,\\cdot\\,,t)$ for all $t \\in \\ensuremath{\\mathbb{C}}$.\n    Let $\\eta$ be this generic $\\ensuremath{\\mathbb{C}}^*$-zero count.\n    Since $\\overline{w}_i$ and $a_{ij}$ are chosen generically,\n    and $\\colv{h}(\\,\\cdot\\,,1) \\equiv \\colv{f}_G(\\;\\cdot\\;)$,\n    we can conclude that  $t = 1$ is outside $Q$ and thus\n    $\\eta$ agrees with the generic $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$\n    that we aims to establish.\n    That is, it is sufficient to show $\\eta \\ge \\nvol(\\ensuremath{\\check{\\nabla}}_G)$. \n\n    Taking a constructive approach,\n    we now construct $\\eta$ smooth curves (of one real-dimension),\n    that will connect the $\\ensuremath{\\mathbb{C}}^*$-zeros of $\\colv{f}_G$\n    and the collection of $\\ensuremath{\\mathbb{C}}^*$-zeros of the special systems described in \\Cref{lem: tree system}.\n    This connection allows us to count the number of curves.\n    \n    Along a ray $t : (0,1] \\to \\ensuremath{\\mathbb{C}}$ on the complex plane parametrized by\n    $\n        t(\\tau) = \\tau e^{i \\theta}\n    $\n    for a choice of $\\theta \\in [0,2\\pi)$ that avoids $Q$\n    (i.e., $t(\\tau) \\not\\in Q$ for all $\\tau \\in (0,1]$),\n    the function $\\colv{h}$ is given by\n    \\[\n        h_i (x_1,\\ldots,x_n,t) =\n        h_i (x_1,\\ldots,x_n, \\tau e^{i \\theta}) =\n        \\overline{w}_i - \\sum_{j \\in \\ensuremath{\\mathcal{N}}_G(i)}\n        a_{ij} e^{i \\theta \\omega_{ij}}\n        \\tau^{\\omega_{ij}}\n        \\left(\n            \\frac{x_i}{x_j} -\n            \\frac{x_j}{x_i}\n        \\right),\n    \\]\n    and $\\colv{h}$  has $\\eta$ isolated and regular $\\ensuremath{\\mathbb{C}}^*$-zeros\n    for all $\\tau \\in (0,1]$.\n    By the principle of homotopy continuation,\n    the zero set of $\\colv{h}$ form $\\eta$ smooth curves in $(\\ensuremath{\\mathbb{C}}^*)^n \\times (0,1]$\n    smoothly parametrizable by $\\tau$.\n    The problem is now reduced to counting these curves.\n\n    Fixing such a curve $C$, the asymptotic behavior of $C$ as $\\tau \\to 0$,\n    in a compactification of $(\\ensuremath{\\mathbb{C}}^*)^n$ can be characterized by\n    the solutions to an initial system of $\\colv{h}$,\n    as a system of Laurent polynomials in \n    $\\ensuremath{\\mathbb{C}}\\{\\tau\\}[x_1^{\\pm 1}, \\ldots, x_n^{\\pm 1}]$.\n    Stated in affine coordinates, by the Fundamental Theorem of Tropical Geometry,\n    there exists a system of convergent Puiseux series\n    $x_1(\\tau),\\ldots,x_n(\\tau) \\in \\ensuremath{\\mathbb{C}}\\{\\tau\\}$\n    that represents the germ of $C$, as an analytic variety at $\\tau=0$,\n    such that the leading coefficients, as a point in $(\\mathbb{C}^*)^n$ satisfies\n    the initial system $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(\\colv{h})$,\\footnote{%\n        We do not claim $\\colv{h}$ form a tropical basis,\n        nor do we require that $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(\\colv{h})$ generate the corresponding initial ideal.\n        Yet, as we will show, an initial system of $\\colv{h}$\n        of the special form we will describe\n        is sufficient to uniquely determine the leading coefficients\n        of the Puiseux series expansion of a solution path.\n    }\n    and $\\rowv{v} = (v_1,\\ldots,v_n) \\in \\mathbb{Q}^n$\n    are the orders of the Puiseux series\n    $x_1(\\tau),\\ldots,x_n(\\tau) \\in \\ensuremath{\\mathbb{C}}\\{\\tau\\}$.\n    Therefore, it is sufficient to show that\n    there are at least $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$ distinct initial systems of $\\colv{h}$\n    that will each contribute one solution path.\n\n    By \\Cref{lem: generic lifting},\n    the regular subdivision $\\Delta_\\omega = \\Sigma_\\omega(\\ensuremath{\\check{\\nabla}}_G)$\n    is a unimodular triangulation.\n    Fix a simplex $\\Delta \\in \\Delta_\\omega$,\n    and let $(\\rowv{v},1)$ be the upward pointing inner normal vector\n    that defines $\\Delta$,\n    then by construction, there is an affinely independent set\n    $\\{\\colv{a}_1,\\ldots,\\colv{a}_n\\} \\subset \\partial \\ensuremath{\\check{\\nabla}}_G$\n    such that\n    \\begin{align*}\n        \\inner{ \\rowv{v} }{ \\colv{a}_k } + \\omega(\\colv{a}_k) &= 0\n        \\quad\\text{for } k = 1,\\ldots,n,\\; and\n        \\\\\n        \\inner{ \\rowv{v}   }{ \\colv{a} } + \\omega(\\colv{a}) &> 0\n        \\quad\\text{for all } \\colv{a} \\in \\ensuremath{\\check{\\nabla}}_G \\setminus \\{ \\colv{0}, \\colv{a}_1,\\ldots,\\colv{a}_n \\}.\n    \\end{align*}\n    Therefore, the exponent vectors associated with monomials with\n    nonzero coefficients in $\\ensuremath{\\operatorname{init}}_{\\rowv{v}} (\\colv{h})$ are exactly\n    $\\{ 0, \\colv{a}_1,\\ldots,\\colv{a}_n \\}$.\n    By \\Cref{lem: simplex tree}, $\\dig{T} = \\dig{G}_\\Delta$ is\n    an acyclic subdigraph and its associated subgraph $T$\n    is a spanning tree of $G$.\n    Indeed, $\\ensuremath{\\operatorname{init}}_{\\rowv{v}} (\\colv{h})$ consists of Laurent polynomials\n    \\[\n        \\overline{w}_i\n        - \\sum_{j \\in \\mathcal{N}^+_{\\vec{T}}(i)} a_{ij} \\frac{x_i}{x_j}\n        + \\sum_{j \\in \\mathcal{N}^-_{\\vec{T}}(i)} a_{ij} \\frac{x_j}{x_i}\n        \\quad\\text{for } i = 1,\\ldots,n.\n    \\]\n    By \\Cref{lem: tree system}, this system has a unique $\\ensuremath{\\mathbb{C}}^*$-zero,\n    which is regular.\n    Following from the principle of homotopy continuation,\n    it gives rise to a smooth curve defined by\n    $\\colv{h}(\\rowv{x},\\tau e^{i\\theta}) = 0$ in $(\\ensuremath{\\mathbb{C}}^*)^n \\times (0,1]$.\n\n    This argument holds for every simplex in $\\Delta_\\omega$.\n    That is, each simplex $\\Delta \\in \\Delta_\\omega$ contributes a curve define by $\\colv{h} = \\colv{0}$.\n    Moreover, these curves are distinct,\n    so the number of curves $\\eta$ satisfies\n    $\n        \\eta \\ge |\\Delta_\\omega| = \\nvol(\\ensuremath{\\check{\\nabla}}_G),\n    $\n    which completes the proof.\n\\end{proof}\n\nThis establishes the fact that\nthe generic $\\ensuremath{\\mathbb{C}}^*$-zero count for the algebraic Kuramoto system $\\colv{f}_G$\nis exactly the adjacency polytope bound,\ndespite the algebraic constraints on the parameters.\nNote that the BKK bound for $\\colv{f}_G$ is always between\nthe generic $\\ensuremath{\\mathbb{C}}^*$-zero count and the adjacency polytope bound.\nTherefore, we can indirectly derive the Bernshtein-genericity\nof $\\colv{f}_G$.\n\n\n\\begin{corollary}\\label{cor:bernshtein_general}\n    For generic real or complex $\\overline{w}_1,\\ldots,\\overline{w}_n$\n    and generic but symmetric real or complex $k_{ij} = k_{ji} \\ne 0$\n    for $\\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G)$,\n    the algebraic Kuramoto system \\eqref{equ: algebraic kuramoto} is Bernshtein-general,\n    and\n    \\[\n        \\mvol(\\ensuremath{\\operatorname{Newt}}(f_{G,1}),\\ldots,\\ensuremath{\\operatorname{Newt}}(f_{G,n})) =\n        \\nvol(\\conv(\\ensuremath{\\operatorname{Newt}}(f_{G,1}) \\cup \\cdots \\cup \\ensuremath{\\operatorname{Newt}}(f_{G,n}))) =\n        \\nvol(\\ensuremath{\\check{\\nabla}}_G).\n    \\]\n\\end{corollary}\n\n\\begin{remark}\n    We remark that the proof of \\Cref{thm: generic root count}\n    utilizes many ideas from the polyhedral homotopy of\n    Huber and Sturmfels \\cite{HuberSturmfels1995Polyhedral}.\n    The main differences are that we consider a lifting function\n    that preserves the relationships among the coupling coefficients\n    and we choose generators of the ideals at ``toric infinity'' with a nice tree structure\n    instead of requiring them to be binomial.\n    \n    Finally, we draw attention to the tropical nature of this proof.\n    The first half of the proof establishes the tropical version\n    of the generic root count result:\n    With the assignment of the valuation\n    $\\operatorname{val}(\\overline{w}_i) = 0$ and generic but symmetric\n    $\\operatorname{val}(a_{ij}) = \\operatorname{val}(a_{ji}) = \\omega_{ij} = \\omega_{ji}$,\n    we showed that the intersection number of the tropical hypersurfaces\n    defined by $f_{G,1},\\ldots,f_{G,n}$ is bounded below by $\\nvol(\\ensuremath{\\check{\\nabla}})$\n    which is also the stable self-intersection number\n    of the tropical hypersurface defined by the randomized Kuramoto system\n    (as defined in \\Cref{sec: randomized}) with the same valuation.\n    Computing generic root count via stable tropical intersection\n    is the topic of a recent paper by\n    Paul Helminck and Yue Ren \\cite{HelminckRen2022Generic},\n    in which the special case for complete networks is studied as an example.\n\\end{remark}\n\n\nAnother consequence of \\Cref{thm: generic root count}\nis the monotonicity of the generic $\\ensuremath{\\mathbb{C}}^*$-zero count,\nsince the volume of the adjacency polytope is strictly increasing\nas new edges and nodes are added.\n\n\\begin{corollary}\n    Let $(G_1,K_1,\\colv{w}_1)$ and $(G_2,K_2,\\colv{w}_2)$ be two\n    connected networks such that $G_1 < G_2$.\n    Then the generic $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_{G_1}$\n    is strictly less than that of $\\colv{f}_{G_2}$.\n\\end{corollary}\n\nIn addition, we also get that the phase-delayed algebraic Kuramoto system and lossy PV-type algebraic power flow system are Bernshtein-general.\n\n\n\\begin{corollary}\\label{cor phase delay}\n    For generic choices of $\\overline{w}_i, a_{ij}, C_{ij}$,\n    the algebraic Kuramoto systems with phase delays\n    \\eqref{equ: delayed Kuramoto}\n    is Bernshtein-general,\n    and its $\\ensuremath{\\mathbb{C}}^*$-zero count equals to the normalized volume of $\\nabla_G$.\n\\end{corollary}\n\\begin{proof}\n    As noted before, for generic choices of $\\overline{w}_1,\\ldots,\\overline{w}_n$,\n    all $\\ensuremath{\\mathbb{C}}^*$-zeros of this system are isolated and regular.\n    Let $\\eta$ be the generic number of $\\ensuremath{\\mathbb{C}}^*$-zeros this system has.\n    By \\Cref{thm:bernshtein-b} and \\Cref{cor:bernshtein_general},\n    we know $\\eta \\leq \\nvol(\\ensuremath{\\check{\\nabla}}_G)$.\n    By the Parameter Homotopy Theorem \\cite{MorganSommese1989Coefficient},\n    the number of $\\ensuremath{\\mathbb{C}}^*$-zeros of this parametrized system constant on\n    a nonempty Zariski open set of the parameters.\n    By setting $C_{ij} = 1$, we recover \\eqref{equ: algebraic kuramoto}\n    and by \\Cref{thm: generic root count} and \\Cref{cor:bernshtein_general},\n    this system is Bernshtein-general,\n    with $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$ $\\ensuremath{\\mathbb{C}}^*$-zeros.\n    This shows that $\\eta \\geq \\nvol(\\ensuremath{\\check{\\nabla}}_G)$, giving the result.\n\\end{proof}\n\nFrom the same argument, we get the generic zero count \nfor the PV-type power flow system.\n\n\\begin{corollary}\n    For generic choices of $P_i, b_{ij},g_{ij}$,\n    the lossy PV-type algebraic power flow system \\eqref{eq:pfeqs}\n    is Bernshtein-general,\n    and its $\\ensuremath{\\mathbb{C}}^*$-zero count equals to the normalized volume of $\\ensuremath{\\check{\\nabla}}_G$.\n\\end{corollary}\n\n\\section{Explicit genericity conditions on coupling coefficients and refined zero count}\n\\label{sec: explicit genericity conditions}\n\nSo far\nwe have established that the generic $\\ensuremath{\\mathbb{C}}^*$-zero count of\nthe algebraic Kuramoto system $\\colv{f}_G$\nis the adjacency polytope bound $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$.\nThat is, for almost all choices of the parameters\n$\\vec{w}$ (natural frequencies) and $K$ (coupling coefficients),\nthe $\\ensuremath{\\mathbb{C}}^*$-zero count for $\\colv{f}_G$ equals $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$.\n\n\nThis section aims to understand when\nthe $\\ensuremath{\\mathbb{C}}^*$-zero count drops below the generic root count and by how much.\nWe focus on the effect of coupling coefficients $\\{ k_{ij} \\}$\nwhile leaving the choices of the natural frequencies $\\colv{w}$ generic,\nwhich ensures all $\\ensuremath{\\mathbb{C}}^*$-zeros of $\\colv{f}_G$ and $\\colv{f}^*_G$ are isolated and regular.\n\n\n\\begin{definition}\\label{def: global exceptional}\n    Given a nontrivial and connected graph $G$ with $n+1$ nodes,\n    we define its \\term{exceptional coupling coefficients}\n    $\\mathcal{K}(G)$ to be the set of\n    symmetric and nonzero coupling coefficients $K = \\{ k_{ij} \\}$\n    such that the number of isolated $\\ensuremath{\\mathbb{C}}^*$-zeros of  $\\colv{f}_{(G,K,\\colv{w})} =0$\n     is strictly less than $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$,\n    for any choice of $\\colv{w} \\in \\ensuremath{\\mathbb{C}}^n$.\n\\end{definition}\n\n\nWe first give an \nalgebraic and graph-theoretic description for $\\mathcal{K}(G)$.\nThen, for certain families of graphs,\nwe decompose $\\mathcal{K}(G)$ into strata,\non which we develop refined root counts.\n\n\\subsection{Exceptional coupling coefficients for a facial systems}\\label{sec: bad coupling}\n\nWe start with an analysis of a single facial system.\nFor a given face $F$ of $\\ensuremath{\\check{\\nabla}}_G$,\nwe describe the coupling coefficients\nfor which the facial system $\\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)$ has a $\\ensuremath{\\mathbb{C}}^*$-zero,\n signaling the genericity condition for \\Cref{thm: generic root count} is broken. \n\n\n\\begin{definition}\\label{def: bad coupling}\n    Let $F \\ne \\varnothing$ be a proper face of $\\ensuremath{\\check{\\nabla}}_G$ such that $G_F$ is connected.\n    We define the set \n    \\[\n        \\mathcal{K}(\\dig{G}_F) = \\{\n            \\rowv{k}(\\dig{G}_F) \\in (\\ensuremath{\\mathbb{C}}^*)^{|F|} \\mid\n            \\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G) = \\colv{0} \\ \\text{has a } \\ensuremath{\\mathbb{C}}^*\\text{-solution}\n        \\}\n    \\]\n    to be the set of \\term{exceptional coupling coefficients} with respect to the face $F$\n    (or $\\dig{G}_F$).\n\\end{definition}\n\nRecall that $\\rowv{k}(\\dig{G}_F)$ is the vector whose entries are the\ncoupling coefficients $k_{ij}$ for arcs $(i,j)$ in $\\dig{G}_F$.\nBy Bernshtein's Second Theorem (\\Cref{thm:bernshtein-b}),\nif, for a proper face $F$ of $\\ensuremath{\\check{\\nabla}}_G$,\nthe coupling coefficients $\\rowv{k}(\\dig{G}_F)  \\in \\mathcal{K}(\\dig{G}_F)$,\nthen the algebraic Kuramoto system $\\colv{f}_G$ is not Bernshtein-general,\nand its isolated $\\ensuremath{\\mathbb{C}}^*$-zero count is strictly less than the\nadjacency polytope bound $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$.\nNote that since we fixed a single face,\nthis is inherently a local description of a patch of $\\mathcal{K}(G)$\n(\\Cref{def: global exceptional}).\nBy \\Cref{thm: generic root count,thm:bernshtein-b},\n$\\mathcal{K}(\\dig{G}_F)$ is contained in a proper and Zariski closed subset of Lebesgue measure zero\\footnote{%\n    This result is justified more directly in an unlabelled paragraph\n    in Bernshtein's paper \\cite{Bernshtein1975Number}.\n}.\nIn the following, we describe the structure of $\\mathcal{K}(\\dig{G}_F)$.\n\n\n\nTo lay down the algebraic foundation,\nwe first show that every facial system of $\\colv{f}^*_G$\ncan be transformed into a ``cycle form'',\nwhich can be understood more easily using graph theoretic information.\n\n\n\\begin{lemma}\\label{lem: face system}\n    Let $F \\ne \\varnothing$ be a proper face of $\\ensuremath{\\check{\\nabla}}_G$,\n    $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ and $\\rowv{a}(\\dig{G}_F)$\n    be the corresponding reduced incidence matrix of $\\dig{G}_F$ and its coupling vector, respectively.\n    Then $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$ \n    if and only if\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_F)\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top\n        =\n        \\colv{0}.\n    \\]\n\\end{lemma}\n\nThis form will be referred to as the ``cycle form''\nof the facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$\nsince the null space of the incidence matrix of a digraph\nis spanned by fundamental cycle vectors.\n\n\\begin{proof}\n    Recall that $\\colv{f}^*_G = R \\cdot \\colv{f}_G$\n    for a nonsingular matrix $R = [r_{ij}]$\n    (see \\Cref{sec: randomized}).\n    Since we assumed $a_{ij} = a_{ji}$,\n    for each $k = 1,\\dots,n$, we have\n    \\begin{align*}\n        \\ensuremath{\\operatorname{init}}_F (f^*_{G,k})(\\rowv{x}) \n        &= \n        \\sum_{(i,j) \\in \\dig{G}_F} \n            (r_{ki} - r_{kj}) \\, \n            a_{ij} \\, \\rowv{x}^{\\colv{e}_i - \\colv{e}_j}\n        = \n        \\sum_{(i,j) \\in \\dig{G}_F} \n            \\inner{ \\rowv{r}_k }{ \\colv{e}_i - \\colv{e}_j } \\,\n            a_{ij} \\, \\rowv{x}^{\\colv{e}_i - \\colv{e}_j},\n    \\end{align*}\n    where $\\rowv{r}_k$ is the $k$-th row of the matrix $R$.\n    Since vectors of the form $\\colv{e}_i - \\colv{e}_j$ \n    for $(i,j) \\in \\ensuremath{\\mathcal{E}}(\\dig{G}_F)$\n    are exactly the columns in $\\ensuremath{\\check{Q}}(\\dig{G}_F)$,\n    \\[\n        \\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)(\\rowv{x}) \n        = \n        R \\, \\ensuremath{\\check{Q}}(\\dig{G}_F) \\,\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top.\n    \\]\n    Since the square matrix $R$ is assumed to be nonsingular,\n    this establishes the equivalence.\n\\end{proof}\n\nThis cycle form suggests a strong tie between\nthe topological features of face subgraphs of $G$ and\nthe algebraic features of the initial systems of\nthe randomized Kuramoto system $\\colv{f}^*_G$.\nThe rest of this paper is devoted to exploring this connection\nby leveraging the wealth of existing knowledge about\nthe facial complex of $\\ensuremath{\\check{\\nabla}}_G$.\nIn particular, \nwe will show that the cycle form of facial systems\nproduces concrete genericity conditions on the coupling coefficients.\n\n\n\nNote that a facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$ can always be\ndecomposed into decoupled collections of facial systems\ncorresponding to weakly connected components of $\\dig{G}_F$.\nWe state this as a lemma.\n\n\n\\begin{lemma}\\label{lem: componentwise facial system}\n    For a nonempty and proper face $F$ of $\\ensuremath{\\check{\\nabla}}_G$\n    whose facial subdigraph $\\dig{G}_F$ consists of\n    weakly connected components $\\dig{H}_1,\\ldots,\\dig{H}_\\ell$,\n    there exists nonempty faces $F_1,\\ldots,F_\\ell$ of $F$\n    (which are also faces of $\\ensuremath{\\check{\\nabla}}_G$)\n    such that $\\dig{H}_i = \\dig{G}_{F_i}$ for $i=1,\\ldots,\\ell$\n    and\n    \\[\n        \\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}\n        \\quad\\Longleftrightarrow\\quad\n        \\ensuremath{\\operatorname{init}}_{F_i}(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}\n        \\quad\\text{for each } i = 1,\\ldots,\\ell.\n    \\]\n\\end{lemma}\n\\begin{proof}\n    Let $V_i = \\ensuremath{\\mathcal{V}}(\\dig{H}_i)$ for $i=1,\\ldots,\\ell$ with $0 \\in V_1$.\n    We arrange the coordinates of $\\ensuremath{\\mathbb{Z}}^n$\n    according to the grouping $V_1,\\ldots,V_\\ell$\n    so that an inner normal vector $\\rowv{\\alpha} \\in \\ensuremath{\\mathbb{R}}^n$\n    that defines the face $F$ of $\\ensuremath{\\check{\\nabla}}_G$ is written as\n    $\n        \\rowv{\\alpha} = \n        \\begin{bmatrix}\\,\n            \\rowv{\\alpha}_1 &\n            \\cdots &\n            \\rowv{\\alpha}_\\ell \\,\n        \\end{bmatrix}\n    $\n    where $\\rowv{\\alpha}_i \\in \\ensuremath{\\mathbb{R}}^{|V_i|}$\n    corresponds to nodes in $V_i$.\n    Then for $i=1,\\ldots,\\ell$,\n    \\[\n        \\rowv{v}_i :=\n        \\begin{bmatrix}\n            \\,\n            \\rowv{0}_{ |V_1| + \\cdots + |V_{i-1}| } &\n            \\rowv{\\alpha}_i &\n            \\rowv{0}_{|V_{i+1}| + \\cdots + |V_\\ell|} \\,\n        \\end{bmatrix}\n        \\in \\ensuremath{\\mathbb{R}}^n\n    \\]\n    defines a face $F_i$ of $\\ensuremath{\\check{\\nabla}}_G$\n    such that $\\dig{H}_i = \\dig{G}_{F_i}$.\n    \n    By grouping the rows and columns of $\\ensuremath{\\check{Q}}(\\dig{G}_F)$\n    according to the nodes and arcs in $H_1,\\ldots,H_\\ell$,\n    $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ has the block structure\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_F) =\n        \\left[\n        \\begin{smallmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{H}_1) & &  \\\\\n            & \\ddots & \\\\\n            & & \\ensuremath{\\check{Q}}(\\dig{H}_\\ell)\n        \\end{smallmatrix}\n        \\right].\n    \\]\n    Therefore, the cycle form of $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G) = \\colv{0}$,\n    i.e., \n    $\n        \\ensuremath{\\check{Q}}(\\dig{G}_F)\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top\n        =\n        \\colv{0},\n    $\n    is equivalent to\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{H}_i)\n        \\left( \n            \\rowv{x}_i^{ \\ensuremath{\\check{Q}}(\\dig{H}_i) }\n            \\circ\n            \\rowv{a}(\\dig{H}_i)\n        \\right)^\\top\n        =\n        \\colv{0}\n        \\quad\\text{for each } i = 1,\\ldots,\\ell,\n    \\]\n    where $\\rowv{x}_i$ contain the coordinates corresponding to nodes in $\\dig{H}_i$.\n    This is equivalent to\n    $\\ensuremath{\\operatorname{init}}_{F_i}(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$ for each $i=1,\\ldots,\\ell$,\n    by \\Cref{lem: face system}.\n\\end{proof}\n\nBased on this observation,\nit is sufficient to consider facial systems\ncorresponding to connected facial subgraphs.\nWe now give a description of the coupling coefficients\nfor which the facial system $\\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)$ has a $\\ensuremath{\\mathbb{C}}^*$-zero.\nRecall, a \\emph{bridge} of a graph is an edge\nthat is not contained in any cycle of the graph.\nA graph that contains no bridges is said to be \\emph{bridgeless}.\nWith respect to a spanning tree $T$ of a connected graph,\neach edge outside of $T$ induces a unique cycle in this graph containing this edge. We call the collection of such cycles the \\emph{fundamental cycles} with respect to $T$.\n\n\n\n\\begin{theorem}\\label{thm: facial cycle condition}\n    Consider a corank-$d$ face $F$ of $\\ensuremath{\\check{\\nabla}}_G$ such that $G_F$ is connected.\n    \\begin{itemize}\n        \\item \n        If $G_F$ contains a bridge,\n        then $\\mathcal{K}(\\dig{G}_F) = \\varnothing$.\n        \n        \\item \n        If $G_F$ is bridgeless,\n        for a choice of a spanning tree $T$ of $G_F$,\n        let $C_1,\\ldots,C_d < G_F$ be the fundamental cycles with respect to $T$\n        and let $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d$ be their associated primitive cycle vectors\n        with respect to $\\dig{G}_F$.\n        We define the very affine variety $V(\\dig{G}_F)$ \n        to be the set of $(\\rowv{k},\\rowv{v}) \\in (\\ensuremath{\\mathbb{C}}^*)^{2|F|}$\n        satisfying\n        \\begin{equation*}\n            \\left\\{\n            \\begin{aligned}\n            \\rowv{k}^{\\colv{\\eta}_i} &= \\rowv{v}^{\\colv{\\eta}_i}\n            \\quad\\text{for } i = 1,\\ldots,d.\\\\\n            \\rowv{v} \\, \\ensuremath{\\check{Q}}(\\dig{G}_F)^\\top &= \\rowv{0}.\n            \\end{aligned}\n            \\right.\n        \\end{equation*}\n        Let $\\pi:(\\ensuremath{\\mathbb{C}}^*)^{2|F|} \\to (\\ensuremath{\\mathbb{C}}^*)^{|F|}$ be\n        the projection onto the first $|F|$ coordinates\n        (i.e. the $\\rowv{k}$ coordinates). \n        Then $\\mathcal{K}(\\dig{G}_F) = \\pi(V(\\dig{G}_F))$.\n    \\end{itemize}\n\\end{theorem}\n\\begin{proof}\n    First, we reduce the general statement to the special case\n    in which $G_F$ is spanning.\n    After relabeling the nodes,\n    we can assume $\\ensuremath{\\mathcal{V}}(G_F) = \\{ 0,\\dots,n' \\}$,\n    and $F \\subset \\ensuremath{\\mathbb{R}}^{n'} \\times \\{ \\colv{0} \\}$.\n    If  $G_F$ is not spanning, i.e., $n > n'$, then \n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_F) =\n        \\begin{bmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{G}_{F'}) \\\\\n            \\rowv{0}_{(n-n') \\times |F| }\n        \\end{bmatrix}.\n    \\]\n    Thus the cycle form of the facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$, namely,\n    \\[\n        \\colv{0}\n        =\n        \\ensuremath{\\check{Q}}(\\dig{G}_F)\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top\n        =\n        \\begin{bmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{G}_{F'}) \\\\\n            \\rowv{0}_{(n-n') \\times |F| }\n        \\end{bmatrix}\n        \\left( \n            \\rowv{x}^{ \n                \\left[\n                \\begin{smallmatrix}\n                    \\ensuremath{\\check{Q}}(\\dig{G}_{F'}) \\\\\n                    \\rowv{0}_{(n-n') \\times |F| }\n                \\end{smallmatrix}\n                \\right]\n            }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top\n    \\]\n    is equivalent to\n    \\begin{equation}\\label{equ: projected face system}\n        \\colv{0}\n        =\n        \\ensuremath{\\check{Q}}(\\dig{G}_{F'})\n        \\left( \n            (\\rowv{x}')^{ \\ensuremath{\\check{Q}}(\\dig{G}_{F'}) }\n            \\circ\n            \\rowv{a}(\\dig{G}_{F'})\n        \\right)^\\top\n    \\end{equation}\n    where $\\rowv{x}' = (x_1,\\dots,x_{n'})$\n    with $(x_{n'+1}, \\dots, x_n) \\in (\\ensuremath{\\mathbb{C}}^*)^{n-n'}$ unconstrained.\n    This equation is, in turn, equivalent to the facet system equation\n    $\\ensuremath{\\operatorname{init}}_{F'}(\\colv{f}^*_{G'})(\\rowv{x}') = \\colv{0}$\n    by \\Cref{lem: face system}.\n    Therefore, it is sufficient to assume that $n = n'$,\n    i.e., $G_F$ is spanning.\n\n\n    We now consider the first case: when $G_F$ contains a bridge.\n    Since the null space of $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ is spanned by\n    primitive cycle vectors, every null vector must contain a zero coordinate,\n    i.e., $\\ker(\\dig{G}_F) \\cap (\\ensuremath{\\mathbb{C}}^*)^{|F|} = \\varnothing$.\n    Consequently, $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$,\n    which is equivalent to\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_{F})\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_{F}) }\n            \\circ\n            \\rowv{a}(\\dig{G}_{F'})\n        \\right)^\\top\n        =\n        \\colv{0}\n    \\]\n    has no $\\ensuremath{\\mathbb{C}}^*$-solutions for any choices $\\rowv{a}(\\dig{G}_F) \\in (\\ensuremath{\\mathbb{C}}^*)^{|F|}$.\n    This establishes the first case.\n    \n    For the second case, $G_F \\le G$ spans $G$. \n    Therefore, a spanning tree $T$ of $G_F$ is also a spanning tree of $G$.\n    Let $\\dig{T}$ be the digraph corresponding to $T$ such that $\\dig{T} < = \\dig{G}_F$,\n    and by rearranging the columns, we can write\n    \\begin{align*}\n        \\ensuremath{\\check{Q}}(\\dig{G}_F) &= \n        \\begin{bmatrix} \\ensuremath{\\check{Q}}(\\dig{T}) & \\colv{v}_1 & \\cdots & \\colv{v}_d \\end{bmatrix} \\\\\n        \\rowv{a}(\\dig{G}_F) &=\n        \\begin{bmatrix} \\rowv{a}(\\dig T) & \\alpha_1 & \\cdots & \\alpha_d \\end{bmatrix}\n    \\end{align*}\n    where $\\colv{v}_1,\\ldots,\\colv{v}_d$ are the incidence vectors\n    associated with arcs of $\\dig{G}_F$ that are outside $\\dig{T}$,\n    and $\\alpha_1,\\ldots,\\alpha_d$ are their complex coupling coefficients.\n    Let $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d \\in \\ensuremath{\\mathbb{Z}}^{|F|}$ be the primitive cycle vectors\n    corresponding to the fundamental cycles induced by these arcs\n    with respect to $\\dig{H}$.\n    Since they are only determined up to a choice of sign, we can assume\n    \\[\n        \\colv{\\eta}_i =\n        \\begin{bmatrix}\n            \\colv{\\tau}_i \\\\\n            -\\colv{e}_i\n        \\end{bmatrix}\n        \\quad\\text{for some } \\colv{\\tau}_i \\in \\{-1,0,+1\\}^{n},\n    \\]\n    in which $\\colv{e}_i$ is considered as a vector in $\\ensuremath{\\mathbb{Z}}^d$.\n    In other words, the first $n$ coordinates (i.e., $\\colv{\\tau}_i$) of $\\colv{\\eta}_i$\n    are chosen so that $\\colv{v}_i = \\ensuremath{\\check{Q}}( \\dig T ) \\, \\tau_i$.\n    By \\Cref{lem: primitive null vector}, $\\inner{ \\rowv{1} }{ \\colv{\\eta}_i } = 0$,\n    therefore, $\\inner{ \\rowv{1} }{ \\colv{\\tau}_i } = 1$.\n\n    \n    Suppose $\\rowv{k}(\\dig{G}_F) \\in \\pi( V(\\dig{G}_F) )$,\n    as defined above, we shall show $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$ has a $\\ensuremath{\\mathbb{C}}^*$-zero.\n    By assumption, there exists an\n    $\\rowv{\\eta} = \\begin{bmatrix} \\rowv{\\tau} & u_1 & \\cdots & u_d \\end{bmatrix} \\in (\\ensuremath{\\mathbb{C}}^*)^{n+d}$\n    such that $\\rowv{\\eta}^\\top \\in \\ker \\ensuremath{\\check{Q}}(\\dig{G}_F)$ and \n    \\[\n        \\rowv{k}(\\dig{G}_F)^{ \\colv{\\eta}_i } = \n        (2\\ensuremath{\\mathfrak{i}} \\cdot \\rowv{a}(\\dig{G}_F))^{ \\colv{\\eta}_i } = \n        (2\\ensuremath{\\mathfrak{i}})^{\\inner{ \\rowv{1} }{ \\colv{\\eta}_i }} \\circ \\rowv{a}(\\dig{G}_F)^{ \\colv{\\eta}_i } = \n        \\rowv{a}(\\dig{G}_F)^{ \\colv{\\eta}_i } = \n        \\rowv{\\eta}^{ \\colv{\\eta}_i }\n        \\quad\\text{for } i = 1, \\ldots,d.\n    \\]\nNote that with this partition of entries,\n    $\\rowv{\\eta}^{\\colv{\\eta}_i} = \\rowv{\\tau}^{ \\colv{\\tau}_i } \\cdot u_i^{-1}$.\n    \n    For any $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$, define \n    \\[\n        \\rowv{x} =\n        (\\lambda \\cdot \\rowv{\\tau} \\circ \\rowv{a}(\\dig T)^{-I})^{ \\ensuremath{\\check{Q}}(\\dig T)^{-1} }\n        \\in (\\ensuremath{\\mathbb{C}}^*)^{n}.\n    \\]\n    Then from a straightforward calculation, we can verify that \n    \\begin{align*}\n        \\rowv{x}^{ \\colv{v}_i }\n        &=\n        \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig T) \\colv{\\tau}_i }\n        =\n        (\\lambda \\cdot \\rowv{\\tau} \\circ \\rowv{a}(\\dig T)^{-I})^{ \\colv{\\tau}_i }.\n        =\n        \\lambda \\cdot \\rowv{\\tau}^{ \\colv{\\tau}_i } \\circ \\rowv{a}(\\dig T)^{-\\colv{\\tau}_i }.\n        =\n        \\lambda \\, u_i \\cdot \\rowv{\\eta}^{ \\colv{\\eta}_i } \\circ \\rowv{a}(\\dig T)^{-\\colv{\\tau}_i },\n    \\end{align*}\n    and thus\n    \\begin{align*}\n        \\rowv{x}^{\\ensuremath{\\check{Q}}( \\dig{G}_F )} \\circ \\rowv{a}(\\dig{G}_F) &=\n        \\begin{bmatrix}\n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig T) } \\circ \\rowv{a}(\\dig T) &\n            \\rowv{x}^{ \\colv{v}_1 } \\cdot \\alpha_1 &\n            \\cdots &\n            \\rowv{x}^{ \\colv{v}_d } \\cdot \\alpha_d\n        \\end{bmatrix}\n        =\n        \\lambda \\cdot \n        \\begin{bmatrix}\n            \\rowv{\\tau} &\n            u_1 &\n            \\cdots &\n            u_d \n        \\end{bmatrix}\n        =\n        \\lambda \\, \\rowv{\\eta}. \n    \\end{align*}\n    That is, $\\ensuremath{\\check{Q}}(\\dig{G}_F) ( \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) } \\circ \\rowv{a}(\\dig{G}_F) )^\\top = \\colv{0}$.\n    By \\Cref{lem: face system}, the facial system $\\ensuremath{\\operatorname{init}}_F \\colv{f}^*_G$\n    has a $\\ensuremath{\\mathbb{C}}^*$-zero.\n    \n    Conversely, suppose $\\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)$ has a zero $\\rowv{x} \\in (\\ensuremath{\\mathbb{C}}^*)^n$.\n    By \\Cref{lem: face system},\n    \\[\n        \\rowv{\\eta} := (\\rowv{x}^{\\ensuremath{\\check{Q}}(\\dig{G}_F)} \\circ \\rowv{a}(\\dig{G}_F))^\\top\n        \\in \\ker \\ensuremath{\\check{Q}}(\\dig{G}_F) \\cap (\\ensuremath{\\mathbb{C}}^*)^{|F|}.\n    \\]\n    Since $\\ker \\ensuremath{\\check{Q}}(\\dig{G}_F)$ is spanned by $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d$,\n    $\\rowv{\\eta} = \\sum_{j=1}^d \\lambda_j \\colv{\\eta}_j^\\top$\n    for some $\\lambda_1,\\ldots,\\lambda_d \\in \\ensuremath{\\mathbb{C}}$.\n    This means \n    \\[\n        \\rowv{\\eta}^{\\colv{\\eta_i}} = \n        (\\rowv{x}^{\\ensuremath{\\check{Q}}(\\dig{G}_F)} \\circ \\rowv{a}(\\dig{G}_F))^{\\colv{\\eta_i}} =\n        \\rowv{x}^{\\ensuremath{\\check{Q}}(\\dig{G}_F)\\colv{\\eta_i}} \\circ \\rowv{a}(\\dig{G}_F)^{\\colv{\\eta_i}} = \n        \\rowv{x}^{\\colv{0}} \\circ \\rowv{a}(\\dig{G}_F)^{\\colv{\\eta_i}} = \n        \\rowv{a}(\\dig{G}_F)^{\\colv{\\eta_i}}\n    \\] \n    for $i=1,\\ldots,d$, since $\\colv{\\eta}_i \\in \\ker \\ensuremath{\\check{Q}}(\\dig{G}_F)$.\n    Also recall that $\\rowv{a}(\\dig{G}_F) = \\frac{1}{2 \\ensuremath{\\mathfrak{i}}}\\rowv{k}(\\dig{G}_F)$\n    and $\\inner{\\mathbf{1}}{\\dig{\\eta_i}} = 0$ for all $i =1,\\ldots,d$, so\n    $\\rowv{a}(\\dig{G}_F)^{\\dig{\\eta}_i} = \\rowv{k}(\\dig{G}_F)^{\\dig{\\eta}_i}$ giving the result.\n\\end{proof}\n\n\n\n\n\\begin{remark}\\label{rmk: C*-orbit}\n    This proof provides an explicit construction of an orbit\n    in the $\\ensuremath{\\mathbb{C}}^*$-zero set of the facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$\n    from which \\Cref{sec: +dimensional} will derive  interesting constructions.\n\\end{remark}\n\n\n\\begin{remark}\\label{rmk: exceptional parametrization}\n    Since, in the context of \\Cref{thm: facial cycle condition},\n    the null space of $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ is spanned by\n    the primitive cycle vectors $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d$,\n    the set $V(\\dig{G}_F)$ can also be more conveniently be defined by\n    \\begin{align*}\n        \\rowv{k}^{\\colv{\\eta}_i} &= \n        \\left(\n            \\sum_{j = 1}^d \\lambda_j \\colv{\\eta}_j^\\top\n        \\right)^{\\colv{\\eta}_i}\n        \\quad\\text{for } i = 1,\\ldots,d\n        &&\\text{and} &\n        \\sum_{j = 1}^d \\lambda_j \\colv{\\eta}_j^\\top &\\in (\\ensuremath{\\mathbb{C}}^*)^{|F|}\n    \\end{align*}\n    with $\\lambda_1,\\ldots,\\lambda_d \\in \\ensuremath{\\mathbb{C}}$\n    being part of the coordinates.\n    We will use the two versions interchangeably.\n\\end{remark}\n\n\n\nThe complexity of the description of $\\mathcal{K}(G_F)$\namounts to the complexity of computing the projection $\\pi(V(\\dig{G}_F))$.\nWe now describe cases where this projection can be computed explicitly.\n    \nA corank-0 face $F$ of $\\ensuremath{\\check{\\nabla}}_G$ is a simplicial face,\ni.e., an affinely independent face.\nIts corresponding facial subgraph $G_F$ must be a forest\nand hence not bridgeless.\n\n\\begin{corollary}\\label{cor: corank-0 face root}\n    For a proper and nonempty corank-0 face $F$ of $\\ensuremath{\\check{\\nabla}}_G$,\n    $\\mathcal{K}(\\dig{G}_F) = \\varnothing$,\n    and the corresponding facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$ has no $\\ensuremath{\\mathbb{C}}^*$-zero\n    for any choice of (nonzero) coupling coefficients. \n    \\qed\n\\end{corollary}\n\nWe now consider the case of a facial subgraph\nconsisting of $d$ cycles sharing a single edge. \n\n\n\\begin{proposition}\\label{prop: shared edge}\n    For an integer $d \\ge 1$,\n    let $F$ be a proper and nonempty corank-$d$ face of $\\ensuremath{\\check{\\nabla}}_G$\n    for which $G_F$ consists of of $d$ cycles\n    $\\{C_1,\\ldots,C_d\\}$ that share exactly one edge $e$\n    (i.e., $\\ensuremath{\\mathcal{E}}(C_i) \\cap \\ensuremath{\\mathcal{E}}(C_j) = \\{ e \\}$ for any pair of distinct $i,j$),\n    and let $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d \\in \\{-1,0,1\\}^{|F|}$\n    be the primitive cycle vectors of $\\dig{G}_F$ corresponding to $C_1,\\ldots,C_d$\n    with consistent signs.\n    Then $\\ensuremath{\\operatorname{init}}_F(\\dig{f}^*)$ has a $\\ensuremath{\\mathbb{C}}^*$-zero if and only if \n    \\begin{equation}\\label{equ: d-cycle facial condition}\n        (-1)^{|C_1|\/2} \\rowv{k}(\\dig{G}_F)^{\\colv{\\eta}_1} \n        \\,+\\, \\cdots \\,+\\,\n        (-1)^{|C_d|\/2} \\rowv{k}(\\dig{G}_F)^{\\colv{\\eta}_d}\n        = 1.\n    \\end{equation}\n\\end{proposition}\n\nHere, the ``consistent'' sign assignment means\nthe common nonzero entries in $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d$\nall share the same sign\nfor $d > 1$. \nIf $d = 1$, the set $\\{ \\colv{\\eta}_1 \\}$\nthat contains the only choice of primitive cycle vector,\nregardless of the signs, will still be considered as consistent for simplicity.\n\n\\begin{proof}\n    Recall that the face subgraph $G_F$ must be bipartite\n    (\\Cref{thm: faces are max bipartite})\n    and therefore the cycles $C_1,\\ldots,C_d$ must be even.\n    For $i=1,\\ldots,d$,\n    let $m_i = |C_i|$, $T_i = C_i - \\{ e \\}$,\n    and $\\dig{T}_i < \\dig{G}_F$ be the corresponding subdigraph.\n    We order the columns in the reduced incidence matrix $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ so that\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_F) = \n        \\begin{bmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{T}_1) &\n            \\cdots &\n            \\ensuremath{\\check{Q}}(\\dig{T}_d) &\n            \\colv{v}_e\n        \\end{bmatrix},\n    \\]\n    where $\\colv{v}_e$ is the incidence vector associated with the edge $e$ in $\\dig{G}_F$.\n    Since $e$ is contained $C_1,\\ldots,C_d$,\n    there is a unique vector $\\colv{\\tau}_i$, for each $i=1,\\dots,d$,\n    such that $\\colv{v}_e = \\ensuremath{\\check{Q}}(\\dig{T}_i) \\, \\colv{\\tau}_i$.\n    Since $\\operatorname{corank}(F) = d$,\n    $\\ker \\ensuremath{\\check{Q}}(\\dig{G}_F)$ is spanned by the $d$\n    primitive cycle vectors (expressed as row vectors) \n    \\[\n        \\rowv{\\eta}_i \\;= \n        \\begin{tikzpicture}[decoration=brace,baseline={([yshift=-2ex]current bounding box.center)}]\n            \\matrix (m) [matrix of math nodes,left delimiter=[,right delimiter={]}] {\n                0 & \\cdots & 0 &\n                \\colv{\\tau}_i^\\top &\n                0 & \\cdots & 0 &\n                -1 \\\\\n            };\n            \\draw[decorate,thick]\n            (m-1-1.north west) -- node[above=2pt] {\\scriptsize $\\sum_{j=1}^{i-1} m_j$} (m-1-3.north east);\n            \\draw[decorate,thick]\n            (m-1-5.north west) -- node[above=2pt] {\\scriptsize $\\sum_{j=i+1}^d m_j$} (m-1-7.north east);\n        \\end{tikzpicture}\n        \\hspace{8ex} \\text{for } i = 1,\\ldots,d\n    \\]\n    corresponding to the cycles $C_1,\\ldots,C_d$.\n    Recall that, by \\Cref{lem: primitive null vector},\n    $\\inner{ \\rowv{1} }{ \\colv{\\eta}_i } = 0$, and thus\n    there must be $m_i\/2$ positive entries and $m_i\/2 - 1$ negative entries in $\\colv{\\tau}_i$.\n    That is, $(\\colv{\\tau}_i^\\top)^{\\colv{\\tau}_i} = (-1)^{m_i\/2 - 1}$.\n    \n    Then any $\\rowv{v}^\\top \\in \\ker \\ensuremath{\\check{Q}}(\\dig{G}_F) \\cap (\\ensuremath{\\mathbb{C}}^*)^{|F|}$\n    can be expressed as\n    \\[\n        \\rowv{v} = \n        \\sum_{i=1}^d \\lambda_i \\rowv{\\eta}_i = \n        \\begin{bmatrix}\n            \\lambda_1 \\colv{\\tau}_1^\\top &\n            \\cdots &\n            \\lambda_d \\colv{\\tau}_d^\\top &\n            -(\\lambda_1 + \\cdots + \\lambda_d)\n        \\end{bmatrix}\n    \\]\n    for some $\\lambda_i \\in \\ensuremath{\\mathbb{C}}^*$ such that $\\lambda_1 + \\cdots + \\lambda_d \\ne 0$.\n    In this case, it is easy to verify that\n    \\[\n        \\rowv{v}^{\\colv{\\eta}_i} =\n        \\frac{ (-1)^{m_i\/2} \\lambda_i }{ \\lambda_1 + \\cdots + \\lambda_d }\n    \\]\n    \n    \n    By \\Cref{rmk: exceptional parametrization},\n    the exceptional coupling coefficients $\\rowv{k} := \\rowv{k}(\\dig{G}_F)$ \n    are defined by the equations\n    \\begin{equation}\\label{equ: k-lambda}\n        (-1)^{m_i\/2} \\, \\rowv{k}^{\\colv{\\eta}_i} = \\frac{\\lambda_i}{\\lambda_1 + \\ldots + \\lambda_d}\n        \\quad\\quad\\text{for } i = 1,\\ldots,d,\n    \\end{equation}\n    and  $\\mathcal{K}(\\dig{G}_F)$ is the projection of the solution set\n    of this system onto the $\\rowv{k}$ coordinates.\n    It suffices to show that this projection is exactly the hypersurface, $\\mathcal{H}$,\n    defined by \\eqref{equ: d-cycle facial condition}.\n\n\n    Summing equations \\eqref{equ: k-lambda} for $i=1,\\ldots,d$,\n    we get the equation \\eqref{equ: d-cycle facial condition} and therefore, $\\mathcal{K}(\\dig{G}_F) \\subseteq \\mathcal{H}$. \n    Conversely, for any $\\rowv{k} \\in \\mathcal{H}$,\n    let $\\lambda_i = (-1)^{m_i\/2} \\rowv{k}^{\\colv{\\eta}_i} \\in \\ensuremath{\\mathbb{C}}^*$ for $i=1,\\ldots,d$.\n    Then $\\lambda_1 + \\cdots + \\lambda_d = 1 \\ne 0$,\n    and $(-1)^{m_i\/2} \\rowv{k}^{\\colv{\\eta}_i} = \\frac{ \\lambda_i }{ \\lambda_1 + \\cdots + \\lambda_d }$,\n    satisfying \\eqref{equ: k-lambda} so $\\rowv{k} \\in \\mathcal{K}(\\dig{G}_F)$.\n\\end{proof}\n\nIn the special case of $d=1$\n(i.e. $G_F$ contains a unique cycle)\nthe above proposition can be reduced to\na particular simple binomial description for\n$\\mathcal{K}(\\dig{G}_F)$.\n\n\\begin{corollary}\\label{cor: corank-1 face root}\n    Let $F$ be a corank-1 face of $\\ensuremath{\\check{\\nabla}}_G$\n    for which $G_F$ is connected,\n    $\\colv{\\eta}$ be a primitive cycle vector of $\\dig{G}_F$,\n    and $\\rowv{k}(\\dig{G}_F)$ be the corresponding coupling vector.\n    The facial system $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$ has a $\\ensuremath{\\mathbb{C}}^*$-zero if and only if\n    $G_F$ is a cycle and\n    \\begin{equation}\\label{equ: coupling condition for face}\n        \\rowv{k}(\\dig{G}_F)^{\\colv{\\eta}} =\n        (-1)^{|F|\/2}\n    \\end{equation}\n\\end{corollary}\n\n\n\\subsection{Global description of exceptional coupling coefficients}\n\nThe set $\\mathcal{K}(\\dig{G}_F)$\ncharacterizes the coupling coefficients\nthat will cause a specific facial system of $\\colv{f}^*_G$ to have $\\ensuremath{\\mathbb{C}}^*$-zero.\nBy Bernshtein's Second Theorem (\\Cref{thm:bernshtein-b}),\nthe $\\ensuremath{\\mathbb{C}}^*$-root count of $\\colv{f}^*_G$ drops below\nthe adjacency polytope bound\nif and only if the coupling coefficients are in\n $\\mathcal{K}(\\dig{G}_F)$\nfor some nonempty and proper face $F$ of $\\ensuremath{\\check{\\nabla}}_G$.\nBy taking the union of all such faces, this gives a global description of the\nexceptional coupling coefficients $\\mathcal{K}(G)$\nfor which the genericity condition in \\Cref{thm: generic root count} is broken.\n\n\nNote, however, that most faces have no contribution to the set of\nexceptional coupling coefficients.\nCombining \\Cref{thm: faces are max bipartite} and \\Cref{thm: facial cycle condition},\nwe get the following topological constraints on the faces with\npotentially nontrivial contribution to $\\mathcal{K}(G)$.\n\n\\begin{proposition}\\label{prop: global exceptional coupling}\n    Let $\\Phi$ be the subset of nontrivial and proper faces of $\\ensuremath{\\check{\\nabla}}_G$ such that their facial subgraphs are\n    \\begin{enumerate}\n        \\item bridgeless (and hence cyclic), and\n        \\item maximally bipartite in their induced subgraphs.\n    \\end{enumerate}\n    Then the set of exceptional coupling coefficients is the union\n    \\[\n        \\mathcal{K}(G) = \\bigcup_{F \\in \\Phi} \\mathcal{K}(\\dig{G}_F).\n    \\]\n\\end{proposition}\n\nIn this union, subsets $\\mathcal{K}(\\dig{G}_{F})$\nare assumed to be embedded in the common ambient space\nthat contains all coupling coefficients.\n\nBased on this and the local description from \\Cref{prop: shared edge},\nwe can derive the global description of the set of exceptional coupling coefficients\nfor graphs consisting of multiple cycles sharing a single edge.\nThe geometry of the adjacency polytope for such networks\nwas studied in detail by D'Ali, Delucchi, and Micha\\l{}ek \\cite{DAliDelucchiMichalek2022Many}\nand the closely related work~\\cite{ChenKorchevskaia2019Graph}.\n\nAs noted in the proof of \\Cref{prop: shared edge},\nit is sufficient to focus on even cycles,\nsince odd cycles have no contributions to the exceptional coupling coefficients. \n\n\n\n\\begin{proposition}\n    Let $G$ be a graph consisting of $d$ independent even cycles,\n    $C_1,\\ldots,C_d$ that overlap at a single edge $e$\n    (i.e., $\\ensuremath{\\mathcal{E}}(C_i) \\cap \\ensuremath{\\mathcal{E}}(C_j) = \\{ e \\}$ for any $i \\ne j$).\n    Let $U_i \\subset \\{ -1, 0, +1 \\}^{|\\ensuremath{\\mathcal{E}}(G)|}$, for $i=1,\\ldots,d$\n    be the set of balanced primitive cycle vectors\n    associated with $C_1,\\ldots,C_d$.\n    Then the set of exceptional coupling coefficients $\\mathcal{K}(G)$ is\n    the union of subsets\n    \\begin{equation}\\label{equ: shared edge exceptional}\n        \\left\\{\n            \\rowv{k} \\;:\\;\n            \\sum_{j=1}^d\n            (-1)^{|C_{i_j}| \/ 2} \\rowv{k}^{\\colv{\\eta}_{i_j}}\n            = 1\n            \\text{ where }\n            (\\colv{\\eta}_{i_1},\\ldots,\\colv{\\eta}_{i_t}) \\in \n            U_{i_1} \\times \\cdots \\times U_{i_t}\n            \\text{ is consistent}\n        \\right\\}.\n    \\end{equation}\n\\end{proposition}\n\nHere, \\emph{balanced} means each $\\colv{\\eta}_{i_j}$ satisfies\n$\\inner{ \\rowv{1} }{ \\colv{\\eta}_{i_j} } = 0$\n(has equal numbers of $-1$ and $+1$).\nAnd the tuple $(\\colv{\\eta}_{i_1},\\ldots,\\colv{\\eta}_{i_t})$\nis considered \\emph{consistent} \nif $\\{i_1,\\ldots,i_t\\}$ are distinct\nand for each pair $i_j$ and $i_{j'}$,\nthe common nonzero entries of $\\colv{\\eta}_{i_j}$ and $\\colv{\\eta}_{i_{j'}}$\nhave the same signs, as in \\Cref{prop: shared edge}.\n\n\\begin{proof}\n    By \\Cref{prop: global exceptional coupling},\n    the only contributions to $\\mathcal{K}(G)$ come from\n    nonempty proper faces associated with face subgraphs that\n    are bridgeless and maximally bipartite in their induced subgraphs.\n    Let $\\Phi$ be the set of such faces (as in \\Cref{prop: global exceptional coupling}).\n    In this case, these are face subgraphs consisting of\n    exactly a subset of the cycles $C_1,\\ldots,C_d$.\n    \n    It suffices to show that consistent tuples\n    $(\\colv{\\eta}_{i_1},\\ldots,\\colv{\\eta}_{i_t}) \\in U_{i_1} \\times \\cdots \\times U_{i_t}$\n    produce the primitive cycle vectors for all\n    facial subdigraphs $\\{ \\dig{G}_F \\mid F \\in \\Phi \\}$,\n    since, by \\Cref{prop: shared edge},\n    each subset is simply $\\mathcal{K}(\\dig{G}_F)$ in that case.\n    This is given by \\cite[Theorem 7]{ChenDavisKorchevskaia2022Facets} which says that\n    for each \n    $(\\colv{\\eta}_{i_1},\\ldots,\\colv{\\eta}_{i_t}) \\in U_{i_1} \\times \\cdots \\times U_{i_t}$\n    there exists a corank-$t$ face $F$ of $\\ensuremath{\\check{\\nabla}}_G$ such that\n    $\\colv{\\eta}_{i_1},\\ldots,\\colv{\\eta}_{i_t}$\n    are the primitive cycle vectors of $\\dig{G}_F$, and vice versa.\n    With this,\n    the statement is an implication of\n    \\Cref{prop: shared edge} and \\Cref{prop: global exceptional coupling}.\n\\end{proof}\n\n\n\n\\subsection{Refined root count for unicycle networks}\\label{sec: unicycle root count}\n\n$G$ is a unicycle graph\nif it contains a unique cycle.\nFor such a graph,\n\\Cref{cor: corank-1 face root} can be sharpened into a condition\nthat\ndetects and quantifies the drop in the $\\ensuremath{\\mathbb{C}}^*$-root count of $\\colv{f}_G$ relative to the adjacency polytope bound.\n\nIntuitively speaking, all coupling coefficients in $\\mathcal{K}(G)$ are exceptional,\nin the sense that they cause the $\\ensuremath{\\mathbb{C}}^*$-zero count of the algebraic Kuramoto system\nto drop below the adjacency polytope bound.\nHowever, some are ``more exceptional'' than others.\nIn this section, we develop this idea rigorously.\nWe begin by restating the definition of the exceptional coupling coefficients\nwith respect to a face in terms of network properties\nunder the unicycle assumption.\n\n\\begin{definition}[Balanced subnetwork]\\label{def: balanced subnetwork}\n    For a unicycle graph $G$ that contains a unique cycle $O$,\n    let $\\dig{O}^+$ and $\\dig{O}^-$ be digraphs \n    corresponding to assigning clockwise and counterclockwise orientations to $O$, respectively.\n    For an acyclic subdigraph $\\dig{H} < \\dig{G}$, we say $(\\dig{H},K)$ is\n    a \\term{balanced} directed acyclic subnetwork of $(G,K)$ if\n    \\begin{itemize}\n        \\item $| \\ensuremath{\\mathcal{E}}(\\dig{H}) \\cap \\ensuremath{\\mathcal{E}}(\\dig{O}^+) | = | \\ensuremath{\\mathcal{E}}(\\dig{H}) \\cap \\ensuremath{\\mathcal{E}}(\\dig{O}^-) | = | \\ensuremath{\\mathcal{V}}(O) | \/ 2$; and\n        \\item\n            let $\\kappa^+(\\dig{H})$\n            and $\\kappa^-(\\dig{H})$\n            be the products of the coupling coefficients along the arcs in\n            $\\ensuremath{\\mathcal{E}}(\\dig{H}) \\cap \\ensuremath{\\mathcal{E}}(\\dig{O}^+)$ and\n            $\\ensuremath{\\mathcal{E}}(\\dig{H}) \\cap \\ensuremath{\\mathcal{E}}(\\dig{O}^-)$, respectively, then\n            \\[\n                \\frac{\n                    \\kappa^+(\\dig{H})\n                }{\n                    \\kappa^-(\\dig{H})\n                }\n                =\n                (-1)^{|\\ensuremath{\\mathcal{V}}(O)| \/ 2}.\n            \\]\n    \\end{itemize}\n\\end{definition}\n\n\n\n\\begin{wrapfigure}[9]{r}{0.25\\textwidth}\n    \\centering\n    \\captionsetup{width=.24\\textwidth}\n    \\begin{tikzpicture}[\n        main\/.style = {draw,circle},\n        every edge\/.style = {draw,-latex,thick},\n        scale=0.9\n    ] \n        \\node[main] (0) at ( 0, 1) {$0$}; \n        \\node[main] (2) at ( 0,-1) {$2$}; \n        \\node[main] (1) at ( 1.8, 0) {$1$}; \n        \\node[main] (3) at (-1.8, 0) {$3$}; \n        \\path[blue] (1) edge node[above right] {$k_{10}$} (0);\n        \\path[purple] (1) edge node[below right] {$k_{12}$} (2);\n        \\path[purple] (3) edge node[above left] {$k_{30}$} (0);\n        \\path[blue] (3) edge node[below left] {$k_{32}$} (2);\n    \\end{tikzpicture} \n    \\caption{A balanced subnetwork of a 4-cycle}\n    \\label{fig: balanced C4}\n\\end{wrapfigure}\nSince the only type of subnetworks of interest here are directed and acyclic,\nthey will simply be referred to as ``balanced subnetworks''.\nNote that this definition implies that the unique cycle $O$ must be an even cycle.\nFor example, the subdigraph $\\dig{H}$ of $C_4$ shown in \\Cref{fig: balanced C4}\nis acyclic and contains two clockwise arcs $k_{10},k_{32}$\nand two counterclockwise arcs $k_{12},k_{30}$.\nThus,\n$\\kappa^+(\\dig{H}) = k_{10} k_{32}$ and\n$\\kappa^-(\\dig{H}) = k_{12} k_{30}$,\nand it is considered a balanced subnetwork if $k_{10} k_{32} = k_{12} k_{30}$.\nUsing this concept \n\\Cref{cor: corank-1 face root} can be restated into the following.\n\n\n\n\\begin{proposition}\\label{thm: unicycle non-bernshtein condition}\n    Let $(G,K,\\colv{w})$ be a network in which\n    $G$ is a connected unicycle graph.\n    For generic real or complex $\\colv{w}$,\n    the following are equivalent:\n    \\begin{enumerate}\n        \\item the $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$ is strictly less than $\\nvol(\\ensuremath{\\check{\\nabla}}_G)$;\n        \\item $(G,K,\\colv{w})$ has a balanced directed acyclic subnetwork.\n    \\end{enumerate}\n\\end{proposition}\n\n\nIf the unique cycle in $G$ is an odd cycle,\nthen there can be no balanced subnetworks,\nand thus the $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$ must reach\nthe adjacency polytope bound $\\nvol( \\ensuremath{\\check{\\nabla}}_G )$.\nCombined with the known results on adjacency polytope bound\n\\cite{ChenDavisMehta2018Counting},\nwe obtain the following stronger root count.\n\\begin{corollary}\n    Let $(G,K,\\colv{w})$ be a network in which\n    $G$ is a connected unicycle graph\n    and its unique cycle $O$ is an odd cycle.\n    Then for any choices of real or complex $\\colv{w}$ and \n    $K = \\{ k_{ij} \\in \\ensuremath{\\mathbb{C}}^* \\mid \\{ i,j \\} \\in \\ensuremath{\\mathcal{E}}(G) \\}$,\n    the $\\ensuremath{\\mathbb{C}}^*$ zeroes of $\\colv{f}_G$ are all isolated,\n    and the total number, counting multiplicity, equals the adjacency polytope bound\n    \\[\n        \\nvol( \\ensuremath{\\check{\\nabla}}_G ) =\n        2^{\\ell} \\, m \\, \\binom{m-1}{(m-1)\/2},\n    \\]\n    where $m = | \\ensuremath{\\mathcal{V}}(O) |$ and $\\ell = | \\ensuremath{\\mathcal{V}}(G) \\setminus \\ensuremath{\\mathcal{V}}(O) |$.\n    \\qed\n\\end{corollary}\n\n\n\nIf the unique cycle $O$ in $G$ is even,\nthen there may be balanced subnetworks.\nThe usefulness of the concept of balanced subnetworks\nlies in the fact that each maximal balanced subnetwork\nreduces the $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$ by one.\nHere, maximal balanced subnetworks are simply maximal elements\nin the poset of balanced subnetworks.\nTo establish this, we will utilize the facet decomposition homotopy\n\\cite[Theorem 3]{Chen2019Directed},\nwhich we summarize as the following lemma with minor modifications.\n\n\\begin{lemma}[Adapted from Theorem 3 \\cite{Chen2019Directed}]\\label{lem: pyramid system}\n    For generic choices of real or complex $\\colv{w}$\n    and symmetric, nonzero coupling coefficients\n    $K = \\{ k_{ij} \\mid \\{ i, j \\} \\in \\ensuremath{\\mathcal{E}}(G) \\}$,\n    let $\\colv{f}_G$ be the algebraic Kuramoto system derived from $(G,K,\\colv{w})$.\n    Then the number of isolated $\\ensuremath{\\mathbb{C}}^*$-zeros that $\\colv{f}_G$ has equals to \n    the sum over all facets $F \\in \\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G)$ of the $\\ensuremath{\\mathbb{C}}^*$-zero count for pyramid systems of the form\n    \\begin{equation}\\label{equ: pyramid}\n        \\ensuremath{\\check{Q}}(\\dig{G}_F)\n        \\left( \n            \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) }\n            \\circ\n            \\rowv{a}(\\dig{G}_F)\n        \\right)^\\top\n        =\n        \\colv{w}.\n    \\end{equation}\n\\end{lemma}\n\nWe call the above system a \\emph{pyramid system}\\footnote{%\n    In \\cite{Chen2019Directed},\n    the first named author named such a system a ``facet system''.\n    This name, unfortunately, conflicts with the terms facial system and facet system\n    we adopt here.\n}\nbecause the Newton polytope of this system is a pyramid\nof the form $\\conv(\\{ \\colv{0} \\} \\cup F)$.\n\n\n\n\\begin{proposition}\\label{pro: unicycle balanced count}\n    Let $(G,K,\\colv{w})$ be a network in which\n    $G$ is a connected unicycle graph,\n    and its unique $m$-cycle $O$ is an even cycle.\n    Then for generic choices of real or complex $\\colv{w}$,\n    the $\\ensuremath{\\mathbb{C}}^*$-zeros of $\\colv{f}_G$ are all isolated,\n    and the total number equals\n    \\[\n        2^{\\ell} \\,\n        \\left[\n            m \\binom{m-1}{m\/2 - 1} - \\beta \n        \\right],\n    \\]\n    where $\\ell = | \\ensuremath{\\mathcal{V}}(G) \\setminus \\ensuremath{\\mathcal{V}}(O) |$,\n    and $\\beta$ is the number of maximally balanced subnetwork\n   in $(G,K,\\colv{w})$.\n\\end{proposition}\n\n\\begin{proof}\n    By \\Cref{lem: leaf extension}\n    and an induction on leaf nodes,\n    it is sufficient to assume $G = C_N$\n    and show the $\\ensuremath{\\mathbb{C}}^*$-zero count to be\n    $N \\binom{N-1}{N\/2 - 1} - \\beta$.\n    Indeed, it was shown in \\cite[Theorem 13]{ChenDavisMehta2018Counting}\n    that the adjacency polytope bound, i.e., the generic $\\ensuremath{\\mathbb{C}}^*$-zero count\n    is $N \\binom{N-1}{N\/2 - 1}$.\n    So we simply have to show each maximally balanced subnetwork\n    reduces the $\\ensuremath{\\mathbb{C}}^*$-zero count by 1.\n    \n    Fix a facet $F \\in \\mathcal{F}( \\ensuremath{\\check{\\nabla}}_{C_N} )$\n    that corresponds to $\\dig{H} = (\\dig{C}_N)_F$\n    in a maximally balanced subnetwork $(\\dig{H},K,\\colv{w})$.\n    That is, we assume\n    \\[\n        (-1)^{N\/2} = \\frac{ \\kappa^+(\\dig{H}) }{ \\kappa^-(\\dig{H}) } =\n        \\rowv{a}(\\dig{H})^{\\colv{\\eta}},\n    \\]\n    where $\\colv{\\eta}$ is a primitive cycle vector of $\\dig{H}$.\n    We shall first show that under this assumption\n    the number of $\\ensuremath{\\mathbb{C}}^*$-solutions to the pyramid system \\eqref{equ: pyramid} has\n    is exactly $N\/2 - 1$\n    (which is one less than the generic $\\ensuremath{\\mathbb{C}}^*$-zero count of\n    the pyramid system $N\/2$ as shown in\n    \\cite[Proposition 12]{ChenDavisMehta2018Counting}.)\n    \n\n    We start by rewriting \\eqref{equ: pyramid} in a convenient form.\n    Fix an arbitrary arc $e$ of $\\dig{H}$,\n    and let $\\dig{T} = \\dig{H} - \\{ e \\}$,\n    then the corresponding graph $T$ is a spanning tree of $C_N$,\n    and therefore $\\ensuremath{\\check{Q}}(\\dig{T})$ is nonsingular.\n    We shall arrange the entries so that\n    $\\ensuremath{\\check{Q}}(\\dig{H}) = \\begin{bmatrix} \\, \\ensuremath{\\check{Q}}(\\dig{T}) & \\colv{v}_e \\, \\end{bmatrix}$ and\n    $\\rowv{a}(\\dig{H}) = \\begin{bmatrix} \\, \\rowv{a}(\\dig{T}) & a_e \\, \\end{bmatrix}$, and\n    where $\\colv{v}_e$ and $a_e$ are the incidence vector\n    and the complex coupling coefficients for the arc $e$, respectively.\n    By choosing an orientation for the cycle,\n    we can assume the last coordinate of the primitive cycle vector $\\rowv{\\eta}$ is $-1$.\n    That is,\n    $\\rowv{\\eta} = \\begin{bmatrix} \\rowv{\\eta}_{\\dig{T}} & -1 \\end{bmatrix}$,\n    where $\\rowv{\\eta}_{\\dig T}$ is the incidence vector of the oriented cycle\n    with respect to the arcs in $\\dig{T}$.\n    Let $\\rowv{y} = (\\ensuremath{\\check{Q}}(\\dig{T})^{-1} \\colv{w})^\\top$, and $y_{n+1} = 0$,\n    which can be assumed to be generic since $\\colv{w}$ is generic.\n    Then \n    the pyramid system \\eqref{equ: pyramid} is equivalent to\n    \\begin{align}\n        \\rowv{x}^{\\ensuremath{\\check{Q}}(\\dig{T})} \\circ \\rowv{a}(\\dig{T}) &=\n        \\rowv{y} + \\lambda \\rowv{\\eta}_{\\dig{T}}\n        \\label{equ: C tree part}\n        \\\\\n        \\rowv{x}^{\\colv{v}_e} \\cdot a_e &= y_{n+1} - \\lambda.\n        \\label{equ: C extra part}\n    \\end{align}\n    Recall that $\\rowv{a}(\\dig{T}) \\in (\\ensuremath{\\mathbb{C}}^*)^n$,\n    so \\eqref{equ: C tree part} is equivalent to\n    \\[\n        \\rowv{x}^{\\ensuremath{\\check{Q}}(\\dig{T})} =\n        \\rowv{a}(\\dig{T})^{-I} \\circ (\\rowv{y} + \\lambda \\rowv{\\eta}_{\\dig{T}}).\n    \\]\n    Moreover, since $\\ensuremath{\\check{Q}}(\\dig{T})$ is unimodular,\n    it has a unique $\\ensuremath{\\mathbb{C}}^*$-solution for any given $\\lambda \\in \\ensuremath{\\mathbb{C}}$,\n    as long as the right-hand-side is also in $(\\ensuremath{\\mathbb{C}}^*)^n$.\n    That is, under this assumption,\n    the value of $\\rowv{x}$, in a solution,\n    is uniquely determined by the value of $\\lambda$.\n    Therefore, it is sufficient to solve for $\\lambda$\n    and count the distinct solutions.\n    \n    Note that $\\colv{v}_e = \\ensuremath{\\check{Q}}(\\dig{T}) \\, \\colv{\\eta}_{\\dig{T}}$,\n    so\n    substituting \\eqref{equ: C tree part} into \\eqref{equ: C extra part} produces\n    \\[\n        y_{n+1} -\\lambda =\n        \\rowv{x}^{\\colv{v}_e} \\cdot a_e =\n        (\\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{T}) })^{ \\colv{\\eta}_{\\dig{T}} } \\cdot a_e =\n        (\\rowv{a}(\\dig{T})^{-I} \\circ (\\rowv{y} + \\lambda \\rowv{\\eta}_{\\dig{T}}))^{ \\colv{\\eta}_{\\dig{T}} } \\cdot a_e =\n        \\rowv{a}(\\dig{G}_F)^{ - \\colv{\\eta} } \\cdot (\\rowv{y} + \\lambda \\rowv{\\eta}_{\\dig{T}})^{ \\colv{\\eta}_{\\dig{T}} }\n    \\]\n    The problem is now reduced to the problem of counting\n    the number of $\\lambda \\in \\ensuremath{\\mathbb{C}}$ that satisfies this equation.\n    Let $I^+ = \\{ i \\in [n] \\mid \\eta_i = +1 \\}$\n    and $I^- = \\{ i \\in [n] \\mid \\eta_i = -1 \\}$, respectively.\n    Then, the above equation is equivalent to\n    \\[\n        \\prod_{i \\in I^-} (y_i - \\lambda) = \n        \\rowv{a}(\\dig{G}_F)^{ - \\colv{\\eta} } \\prod_{i \\in I^+} (y_i + \\lambda),\n    \\]\n    which is a polynomial equation in $\\lambda$ of degree up to $N\/2$,\n    i.e., up to $N\/2$ solutions for $\\lambda$.\n    However, we will show this upper bound is not attainable.\n    \n\n    Note that the degree $N\/2$ terms of the two sides of the above equation are\n    $(-1)^{N\/2} \\lambda^{N\/2}$ and\n    $\\rowv{a}(\\dig{G}_F)^{ - \\colv{\\eta} } \\lambda^{N\/2}$, respectively.\n    Since we assumed that\n    $\\rowv{a}(\\dig{G}_F)^{ \\colv{\\eta} } = (-1)^{N\/2}$,\n    the degree $N\/2$ terms cancel,\n    and therefore\n    the above equation is actually a polynomial equation\n    in $\\lambda$ of degree up to $N\/2 - 1$\n    with the coefficients being symmetric functions in $\\rowv{y}$,\n    which are assumed to be generic.\n    Consequently, there are exactly $N\/2 - 1$ solutions to this equation in $\\lambda$,\n    which produces $N\/2 - 1$ distinct solutions to the system\n    \\eqref{equ: C tree part}-\\eqref{equ: C extra part}.\n    That is, the pyramid system induced by the facet $F$ has $N\/2 - 1$ distinct $\\ensuremath{\\mathbb{C}}^*$-zeros.\n    \n\n    So far, we have shown that, for any facet associated with\n    a maximally balanced subnetwork, \n    the pyramid system has $N\/2 - 1$ $\\ensuremath{\\mathbb{C}}^*$-solutions,\n    whereas a pyramid system induced by other facets has $N\/2$ $\\ensuremath{\\mathbb{C}}^*$-solutions.\n    It was established in \\cite[Proof of Theorem 13]{ChenDavisMehta2018Counting}\n    that $\\ensuremath{\\check{\\nabla}}_{C_N}$ has $2 \\binom{N-1}{N\/2-1}$ facets.\n    Therefore, the sum of the number of $\\ensuremath{\\mathbb{C}}^*$-solutions\n    to each pyramid system defined in \\eqref{equ: pyramid} is:\n    \\[\n        \\beta \\,\n        \\left(\n            \\frac{N}{2} - 1\n        \\right)\n        + \n        \\left(\n            2 \\binom{N-1}{N\/2-1} - \\beta\n        \\right) \\, \\frac{N}{2}\n        =\n        N \\binom{N-1}{N\/2-1} - \\beta\n        .\n    \\]\n    An application of \\Cref{lem: pyramid system} then gives the result.\n\\end{proof}\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{tikzpicture}[\n        main\/.style = {draw,circle,minimum size=1.5ex},\n        every edge\/.style = {draw,thick,-latex}\n    ] \n        \\node[main] (0) at ( 0, 1) {$0$}; \n        \\node[main] (2) at ( 0,-1) {$2$}; \n        \\node[main] (1) at ( 1.8, 0) {$1$}; \n        \\node[main] (3) at (-1.8, 0) {$3$}; \n        \\path (1) edge node[above right] {$k_{10}$} (0);\n        \\path (1) edge node[below right] {$k_{12}$} (2);\n        \\path (3) edge node[above left] {$k_{30}$} (0);\n        \\path (3) edge node[below left] {$k_{32}$} (2);\n    \\end{tikzpicture} \n    \\hspace{3ex}\n    \\begin{tikzpicture}[\n        main\/.style = {draw,circle,minimum size=1.5ex},\n        every edge\/.style = {draw,thick,-latex}\n    ] \n        \\node[main] (0) at ( 0, 1) {$0$}; \n        \\node[main] (2) at ( 0,-1) {$2$}; \n        \\node[main] (1) at ( 1.8, 0) {$1$}; \n        \\node[main] (3) at (-1.8, 0) {$3$}; \n        \\path (1) edge node[above right] {$k_{10}$} (0);\n        \\path (1) edge node[below right] {$k_{12}$} (2);\n        \\path (0) edge node[above left] {$k_{30}$} (3);\n        \\path (2) edge node[below left] {$k_{32}$} (3);\n    \\end{tikzpicture}\n    \\hspace{3ex}\n    \\begin{tikzpicture}[\n        main\/.style = {draw,circle,minimum size=1.5ex},\n        every edge\/.style = {draw,thick,-latex}\n    ] \n        \\node[main] (0) at ( 0, 1) {$0$}; \n        \\node[main] (2) at ( 0,-1) {$2$}; \n        \\node[main] (1) at ( 1.8, 0) {$1$}; \n        \\node[main] (3) at (-1.8, 0) {$3$}; \n        \\path (0) edge node[above right] {$k_{10}$} (1);\n        \\path (1) edge node[below right] {$k_{12}$} (2);\n        \\path (0) edge node[above left] {$k_{30}$} (3);\n        \\path (3) edge node[below left] {$k_{32}$} (2);\n    \\end{tikzpicture} \n    \\caption{Representatives of the classes of balanced subnetworks in $C_4$}\n    \\label{fig: C4 classes of facet subgraphs}\n\\end{figure}\n\\begin{example}[The 4-cycle case]\\label{ex: C4}\n    Consider the case of $G = C_4$.\n    As shown in \\cite[Theorem 16]{ChenDavisMehta2018Counting}\n    and \\Cref{thm: generic root count},\n    for generic choices of $\\colv{w}$ and $K$,\n    $\\colv{f}_G$ has 12 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n    This is the adjacency polytope bound.\n    Now, with the choice of $\\colv{w}$ remain generic,\n    \\Cref{pro: unicycle balanced count} provides a stratification\n    of the space of coupling coefficients according to the maximum $\\ensuremath{\\mathbb{C}}^*$-zero count.\n    \n    This network has up 6 balanced subnetworks\n    supported by the digraphs shown in \\Cref{fig: C4 classes of facet subgraphs}\n    and their transposes.\n    With these, we have the following decomposition of\n    the coupling coefficients:\n    \\begin{enumerate}[itemsep=1ex]\n        \\item\n            Consider the 1-dimensional ``balancing variety''\n            defined by the binomial system\n            \\[\n                \\left\\{\n                \\begin{aligned}\n                    k_{10} k_{12} k_{32}^{-1} k_{30}^{-1} &= 1 \\\\\n                    k_{10} k_{12}^{-1} k_{32}^{-1} k_{30} &= 1 \\\\\n                    k_{10} k_{12}^{-1} k_{32} k_{30}^{-1} &= 1,\n                \\end{aligned}\n                \\right.\n            \\]\n            which will produce all 6 balanced subnetworks.\n            It can be parametrized by\n            \\[\n                k_{10} = k_{12} = k_{32} = k_{30} = s,\n            \\]\n            for $s \\in \\ensuremath{\\mathbb{C}}^*$.\n            Any choice of $s \\in \\ensuremath{\\mathbb{C}}^*$\n            produces an algebraic Kuramoto system $\\colv{f}_G$\n            with at most $12 - 6 = 6$ $\\ensuremath{\\mathbb{C}}^*$-zeros.\n            For generic $s$,\n            the $\\ensuremath{\\mathbb{C}}^*$-zero count is exactly 6.\n            Moreover, taking $s = 1$\n            and\n            $\\omega_1 = 1.1, \\omega_2 = - 2.1, \\omega_3 = 1$,\n            all six $\\ensuremath{\\mathbb{C}}^*$-zeros can be \\emph{real}\n            (see \\Cref{sec: lemmas}).\n            \n        \\item\n            There are three 2-dimensional balancing varieties\n            defined by two out of three binomial equations\n            from the above system:\n            \\begin{align*}\n                &\n                \\left\\{\n                \\begin{aligned}\n                    k_{10} k_{12} k_{32}^{-1} k_{30}^{-1} &= 1 \\\\\n                    k_{10} k_{12}^{-1} k_{32}^{-1} k_{30} &= 1\n                \\end{aligned}\n                \\right.\n                &&\n                \\left\\{\n                \\begin{aligned}\n                    k_{10} k_{12}^{-1} k_{32}^{-1} k_{30} &= 1 \\\\\n                    k_{10} k_{12}^{-1} k_{32} k_{30}^{-1} &= 1,\n                \\end{aligned}\n                \\right.\n                &&\n                \\left\\{\n                \\begin{aligned}\n                    k_{10} k_{12} k_{32}^{-1} k_{30}^{-1} &= 1 \\\\\n                    k_{10} k_{12}^{-1} k_{32} k_{30}^{-1} &= 1,\n                \\end{aligned}\n                \\right.\n            \\end{align*}\n            each producing 4 balanced subnetworks.\n            They contain choices of coupling coefficients\n            for which $\\colv{f}_G$ has at most 8 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n            A generic choice of coupling coefficients on this variety will produce exactly 8 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n            The first of the three,\n            can be parametrized by\n            \\begin{align*}\n                k_{10} &= s &\n                k_{12} &= t &\n                k_{32} &= \\pm t &\n                k_{30} &= \\pm s ,\n            \\end{align*}\n            where $(s,t) \\in (\\ensuremath{\\mathbb{C}}^*)^2$.\n            The other two,\n            $\\mathcal{K}( \\colv{\\eta}_2 , \\colv{\\eta}_3 )$ and\n            $\\mathcal{K}( \\colv{\\eta}_3 , \\colv{\\eta}_1 )$,\n            can be parametrized similarly.\n            Moreover, taking $s = 1, t = -1.001, \\omega_1 = 1.1, \\omega_2 = -2.1, \\omega_3 = 1$ all $8$ solutions can be real. \n            \n        \\item\n            There are three 3-dimensional balancing varieties\n            defined by binomial equations\n            \\begin{align*}\n                k_{10} k_{12} k_{32}^{-1} k_{30}^{-1} &= 1 &\n                k_{10} k_{12}^{-1} k_{32}^{-1} k_{30} &= 1 &\n                k_{10} k_{12}^{-1} k_{32} k_{30}^{-1} &= 1,\n            \\end{align*}\n            each producing 2 balanced subnetworks.\n            They contain coupling coefficients\n            for which $\\colv{f}_G$ has at most 10 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n            A generic choice of coupling coefficients on one of these varieties gives exactly 10 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n            The first of the three\n            can be parametrized by\n            \\begin{align*}\n                k_{10} &= s &\n                k_{12} &= u\/s &\n                k_{32} &= t &\n                k_{30} &= u\/t ,\n            \\end{align*}\n            for $(s,t,u) \\in (\\ensuremath{\\mathbb{C}}^*)^3$.\n            The other two can be parametrized similarly.\n            Moreover, taking $u = 1.01, s = 1, t = - 1.001, \\omega_1 = 1.1, \\omega_2 = -2.1, \\omega_3 = 1$, all $10$ solutions can be real.\n            \n        \\item\n            The remaining choices of $K$,\n            form a Zariski-dense subset in the space of all\n            coupling coefficients for $C_4$, and\n            define an algebraic Kuramoto system with 12 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n    \\end{enumerate}\n    Under the assumption that $\\colv{w}$ is generic,\n    there are no other possibilities.\n\\end{example}\n\n\\begin{example}[The 6-cycle case]\\label{ex: C6}\n    Similarly, for the case of $G =O= C_6$,\n    it is known that for generic choices of $\\colv{w}$ and $K$,\n    $\\colv{f}_G$ has 60 $\\ensuremath{\\mathbb{C}}^*$-zeros.\n    There are up to 20 balanced subnetworks\n    that form 10 transpose pairs,\n    and they can be described by arc orientation vectors\n    $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_{10}$:\n    \\begin{align*}\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ +1 \\\\ +1 \\\\ -1 \\\\ -1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ +1 \\\\ -1 \\\\ +1 \\\\ -1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ +1 \\\\ -1 \\\\ -1 \\\\ +1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ +1 \\\\ -1 \\\\ -1 \\\\ -1 \\\\ +1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ +1 \\\\ +1 \\\\ -1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ +1 \\\\ -1 \\\\ +1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ +1 \\\\ -1 \\\\ -1 \\\\ +1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ -1 \\\\ +1 \\\\ +1 \\\\ -1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ -1 \\\\ +1 \\\\ -1 \\\\ +1\n        \\end{smallmatrix}\n        \\right]\n        ,&&\n        \\left[\n        \\begin{smallmatrix}\n            +1 \\\\ -1 \\\\ -1 \\\\ -1 \\\\ +1 \\\\ +1\n        \\end{smallmatrix}\n        \\right]\n        ,\n    \\end{align*}\n    where the ordering of the coordinates corresponds to the ordering \n    $\\rowv{k}(O) = [k_{01}, k_{12}, k_{23}, k_{34}, k_{45}, k_{50}]$.\n    The exceptional coupling coefficients for which the $\\ensuremath{\\mathbb{C}}^*$-root count of $\\colv{f}_G$\n    deviates from the generic root count\n    are contained in the zero sets of binomial equations of the form\n    \\[\n        \\rowv{k}(O)^{\\colv{\\eta}_i} = -1,\n    \\]\n    for $i \\in [10]$.\n    This indicates that the exceptional coupling coefficients must\n    include some negative values,\n    which corresponds to repulsive couplings.\n    The full description of the strata of exceptional coupling coefficients will be lengthy,\n    so we only examine two special case.\n    \n    Consider the family of coupling coefficients\n    \\[\n        \\rowv{k}(O) = [ \\pm s, \\ldots, \\pm s ],\n    \\]\n    with an odd number of negative choices, parametrized by $s \\in \\ensuremath{\\mathbb{C}}^*$.\n    Then $\\rowv{k}(O)^{\\colv{\\eta}_i} = -1$ is satisfied for $i=1,\\ldots,10$.\n    By \\Cref{pro: unicycle balanced count},\n    for any $s \\in \\ensuremath{\\mathbb{C}}^*$ and generic choices of $\\colv{w}$,\n    the $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$ is no more than\n    $6 \\cdot \\binom{6-1}{6\/2 - 1} - 2 \\cdot 10 = 40$.\n    This bound is exact for generic choice of $s \\in \\ensuremath{\\mathbb{C}}^*$.\n    \n    Similarly, for the family\n    \\[\n        \\rowv{k}(O) = [ \n            \\pm s, \\pm s,\n            \\pm t, \\ldots, \\pm t\n        ]\n    \\]\n    with an odd number of negative choices,\n    parametrized by $(s,t) \\in (\\ensuremath{\\mathbb{C}}^*)^2$,\n    the conditions $\\rowv{k}(O)^{\\colv{\\eta}_i} = -1$ \n    is satisfied only for $i=5,6,7,8,9,10$.\n    Therefore, the $\\ensuremath{\\mathbb{C}}^*$-zero count of $\\colv{f}_G$ is no more than\n    $6 \\cdot \\binom{6-1}{6\/2 - 1} - 2 \\cdot 6 = 48$\n    for any $(s,t) \\in (\\ensuremath{\\mathbb{C}}^*)^2$ and generic $\\colv{w}$.\n    It is exact for generic choices of $(s,t)$.\n\\end{example}\n\n\n\\section{Positive-dimensional synchronization configurations for bipartite networks}\n\\label{sec: +dimensional}\n\nWe now turn our attention to positive-dimensional zero sets of the algebraic Kuramoto equations.\nThey represent synchronization configurations\nthat have at least one degree of freedom.\n\nWe generalize existing constructions of\npositive-dimensional zero sets for the Kuramoto equations\n\\cite{AshwinBickBurylko2016Identical,coss2018locating,LindbergZachariahBostonLesieutre2022Distribution,sclosa2022kuramoto}\nand characterize conditions under which they arise.\n\n\nThe crucial observation that enables our constructions\nis that when $G$ is bipartite,\nby \\Cref{thm: faces are max bipartite}, $G$ itself is a facet subgraph,\ni.e., there exists a $F \\in \\mathcal{F}(\\ensuremath{\\check{\\nabla}}_G)$ such that $G = G_F$.\nIndeed,\n\\[\n    \\ensuremath{\\check{\\nabla}}_G = \\{ 0 \\} \\cup F \\cup (-F).\n\\]\nThat is, $F$ and $-F$ contains the exponent vectors of all nonconstant terms.\nIf we further assume the oscillators are homogeneous,\ni.e., all the natural frequencies are identical and thus the constant terms\nin $\\colv{f}_G$ are all zero,\nthen the algebraic Kuramoto system can be expressed in the simple form\n\\[\n    \\colv{f}_G(\\rowv{x})\n    = \n    \\ensuremath{\\check{Q}}(G) ( \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{G}_F) } \\circ \\rowv{a}(G))^\\top\n    -\n    \\ensuremath{\\check{Q}}(G) ( \\rowv{x}^{-\\ensuremath{\\check{Q}}(\\dig{G}_F) } \\circ \\rowv{a}(G))^\\top.\n\\]\nNote that both terms are cycle forms of facial systems,\nas shown in \\Cref{lem: face system},\nand they correspond to $\\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)$ and $\\ensuremath{\\operatorname{init}}_{-F} (\\colv{f}^*_G)$,\nrespectively.\nIn the previous section, we already constructed explicit solutions\nto $\\ensuremath{\\operatorname{init}}_F (\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$ under certain restrictions of the coupling coefficients.\nIn the following, we investigate additional restrictions\nthat will ensure the same solution also satisfies\n$\\ensuremath{\\operatorname{init}}_{-F} (\\colv{f}^*_G)(\\rowv{x}) = \\colv{0}$\nand will therefore be a positive dimensional $\\ensuremath{\\mathbb{C}}^*$-zero set to $\\colv{f}_G$ itself.\n\n\n\\subsection{Unicycle graphs}\n\nWe first consider the case when $G$ is a unicycle graph. In this context, \\Cref{thm: unicycle non-bernshtein condition}\ncan be paraphrased as the following necessary condition for the existence\npositive-dimensional $\\ensuremath{\\mathbb{C}}^*$-zero sets.\n\n\\begin{proposition}\\label{thm: unicycle +dimensional necessary}\n    Let $G$ be a (connected) unicycle graph that contains a unique cycle $O$.\n    If the $\\ensuremath{\\mathbb{C}}^*$-zero sets of $\\colv{f}_G$ is positive-dimensional,\n    then $O$ is even and $(G,K)$ contains a balanced subnetwork.\n\\end{proposition}\n\nWe now show that positive dimensional $\\ensuremath{\\mathbb{C}}^*$-zero sets always exist\nfor unicycle networks of homogeneous oscillators that contain a balanced subnetwork.\n\n\\begin{proposition}[Non-isolated $\\ensuremath{\\mathbb{C}}^*$-zero set for unicycle networks]\\label{thm: unicycle +dimensional}\n    Let $G$ be a (connected) unicycle graph on $n+1$ nodes that contains a unique even cycle $O$.\n    Suppose the coupling coefficients satisfy:\n    \\begin{enumerate}\n        \\item $(G,K)$ contains a balanced subnetwork $(\\dig{H},K)$;\n        \\item $\\rowv{k}^2(O) = c^2 \\cdot \\rowv{1}$ for some $c \\in \\ensuremath{\\mathbb{C}}^*$,\n        where $\\rowv{k}^2$ the element-wise square of $\\rowv{k}(O)$.\n    \\end{enumerate}\n    Then for any $w \\in \\ensuremath{\\mathbb{C}}$,\n    the algebraic Kuramoto system $\\colv{f}_G$ derived from homogeneous network $(G,K,w \\, \\colv{1})$\n    has a positive-dimensional $\\ensuremath{\\mathbb{C}}^*$-zero set.\n    Moreover, after relabeling the nodes such that $\\ensuremath{\\mathcal{V}}(O) = \\{ 0,1,\\ldots,m \\}$,\n    the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_G$ contains one-dimensional components\n    parametrized by\n    \\begin{align*}\n        \\begin{bmatrix}\n            x_1(\\lambda) &\n            \\cdots &\n            x_m(\\lambda)\n        \\end{bmatrix}\n        &=\n        (\n            \\lambda \\cdot\n            \\rowv{\\eta}_{\\dig{T}} \\circ\n            (\\rowv{a}(\\dig{T})^{-I})\n        )^{\\ensuremath{\\check{Q}}(\\dig{T})^{-1}}\n        \\\\\n        x_i &= \\pm x_{\\pi(i)}\n        \\quad\\text{for } i = m+1, \\ldots, n,\n    \\end{align*}\n    where $\\dig{T} < \\dig{H}$ such that its corresponding graph $T$ is a spanning tree of $O$,\n    $[ \\rowv{\\eta}_{\\dig{T}} \\; -1 ]$ is a primitive cycle vector of $\\dig{H}$,\n    and $\\pi(i)$ is the tree-order parent of node $i$.\n\\end{proposition}\n\nHere, the choice $w \\, \\colv{1}$ for the coupling coefficients simply means that\nall natural frequencies $w_0,w_1,\\ldots,w_n$ are identical,\ni.e., the network consists of homogeneous oscillators.\nSince the constant terms in the algebraic Kuramoto system \\eqref{equ: algebraic kuramoto}\nare their deviations $\\overline{w}_i = w_i - \\overline{w}$ from the mean,\nthis corresponds to the requirement that all constant terms are zero.\nThe tree-order parent node $\\pi(i)$ of a node $i$ outside the cycle $O$\nis its adjacent node in the unique path to the reference node $0$.\n\n\\begin{proof}\n    By recursive applications of \\Cref{lem: leaf extension},\n    $\\ensuremath{\\mathbb{C}}^*$-zeros for $\\colv{f}_O$ always extend to $\\ensuremath{\\mathbb{C}}^*$-zeros for $\\colv{f}_G$.\n    So without loss, we can assume $G = O,n = m$.\n    Let $F = \\ensuremath{\\check{\\nabla}}_{\\dig{H}}$, then $F$ is a facet of $\\ensuremath{\\check{\\nabla}}_G$,\n    and, \n    \\begin{equation}\\label{equ: p_O}\n        \\colv{f}_G(\\rowv{x})\n        = \n        \\ensuremath{\\check{Q}}( \\dig{H}) ( \\rowv{x}^{ \\ensuremath{\\check{Q}}( \\dig{H}) } \\circ \\rowv{a}( \\dig{H} ))^\\top\n        -\n        \\ensuremath{\\check{Q}}( \\dig{H}) ( \\rowv{x}^{-\\ensuremath{\\check{Q}}( \\dig{H}) } \\circ \\rowv{a}( \\dig{H} ))^\\top\n    \\end{equation}\n    Let $T$ be a spanning tree of $O = G$ \n    and let $\\dig{T}$ be the corresponding subdigraph of $\\dig{H}$.\n   From the proof of \\Cref{thm: facial cycle condition},\n $\\dig{T}$ gives rise to a cocharacter\n    $\\rowv{\\mu} = \\rowv{1} \\ensuremath{\\check{Q}}(\\dig T)^{-1}$.\n    It provides the parametrization \n    \\[\n        \\rowv{x}(\\lambda) =\n        (\\lambda \\cdot \\rowv{\\eta}_{\\dig{T}} \\circ \\rowv{a}(\\dig T)^{-I})^{ \\ensuremath{\\check{Q}}(\\dig T)^{-1} } =\n        (\\rowv{\\eta}_{\\dig{T}} \\circ \\rowv{a}(\\dig T)^{-I})^{ \\ensuremath{\\check{Q}}(\\dig T)^{-1} } \\circ\n        \\lambda^{ \\rowv{\\mu} }\n    \\]\n    of a 1-dimensional orbit in the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\ensuremath{\\operatorname{init}}_F(\\colv{f}^*_G)$\n    and therefore\n    $\\ensuremath{\\check{Q}}( \\dig{H}) ( \\rowv{x}(\\lambda)^{ \\ensuremath{\\check{Q}}( \\dig{H} }) \\circ \\rowv{a}( \\dig{H} ))^\\top = \\colv{0}$.\n    That is, the first term of \\eqref{equ: p_O} vanishes.\n    With a straightforward calculation, we  verify \n    \\begin{align*}\n        \\rowv{x}(\\lambda)^{-\\ensuremath{\\check{Q}}(\\dig{H})} \\circ \\rowv{a}(\\dig{H}) &= \n        (\n            (\n                \\lambda \\cdot\n                \\rowv{\\eta}_{\\dig{T}} \\circ\n                \\rowv{a}(\\dig{T})^{-I}\n            )^{\\ensuremath{\\check{Q}}(\\dig{T})^{-1}}\n        )^{ \\begin{bmatrix} -\\ensuremath{\\check{Q}}(\\dig{T}) & -\\ensuremath{\\check{Q}}(\\dig{T})\\colv{\\eta}_{\\dig{T}} \\end{bmatrix} } \\circ \\rowv{a}(\\dig{H})\n        =\n        - \\frac{ c^2 }{ 4 \\lambda }\n        \\cdot\n        \\rowv{\\eta},\n    \\end{align*}\n    where $c$ is the constant such that $\\rowv{k}^2(\\dig{H}) = c^2 \\cdot \\rowv{1}$\n    from the assumption.\n    Therefore,\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{H})\n        (\n            \\rowv{x}(\\lambda)^{-\\ensuremath{\\check{Q}}(\\dig{H})} \\circ\n            \\rowv{a}(\\dig{H})\n        )^\\top\n        = \\colv{0}.\n    \\]\n    Combined with the calculation above, we see that\n    $\\colv{f}_G(\\rowv{x}(\\lambda)) = \\colv{0}$ for all $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$\n    and thus the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_G = 0$ is positive-dimensional.\n\\end{proof}\n\nThe $\\ensuremath{\\mathbb{C}}^*$-orbit constructed here produces a\npositive-dimensional real zero set to the original transcendental Kuramoto system.\n\n\n\\begin{proposition}[Non-isolated real zero set for unicycle networks]\\label{thm: unicycle real +dimensional}\n    Let $G$ be a (connected) unicycle graph that contains a unique even cycle $O$.\n    Suppose the choice of the coupling coefficients satisfies\n    \\begin{enumerate}\n        \\item $(G,K)$ contains a balanced subnetwork $(\\dig{H},K)$;\n        \\item $\\rowv{k}^2(O) = c^2 \\cdot \\rowv{1}$ for some $c \\in \\ensuremath{\\mathbb{R}}^+$. \n    \\end{enumerate}\n    Then for any $w \\in \\ensuremath{\\mathbb{R}}$,\n    transcendental Kuramoto system \\eqref{equ: kuramoto sin}\n    derived from homogeneous network $(G,K,w \\, \\colv{1})$\n    has positive-dimensional real zero sets.\n\\end{proposition}\n\n\\begin{proof}\n    Let $\\dig{T}$ and $\\rowv{\\eta}_{\\dig{T}}$ be as defined in the above proof.\n    Then for any $\\lambda \\in \\ensuremath{\\mathbb{R}}^{*}$,\n    \\[\n        \\frac{ c e^{\\ensuremath{\\mathfrak{i}} \\lambda} }{2\\ensuremath{\\mathfrak{i}}} \\cdot\n        \\rowv{\\eta}_{\\dig{T}} \\circ\n        \\rowv{a}(\\dig{T})^{-I}\n        =\n        \\frac{ c e^{\\ensuremath{\\mathfrak{i}} \\lambda} }{2\\ensuremath{\\mathfrak{i}}} \\cdot\n        \\rowv{\\eta}_{\\dig{T}} \\circ\n        \\left(\n            \\frac{1}{2 \\ensuremath{\\mathfrak{i}}} \\rowv{k}(\\dig{T})\n        \\right)^{-I}\n        \\in (S^1)^m.\n    \\]\n    Since the group automorphism $\\rowv{x} \\mapsto \\rowv{x}^{ \\ensuremath{\\check{Q}}(\\dig{T})^{-1} }$\n    preserves the the real torus $(S^1)^m \\subset (\\ensuremath{\\mathbb{C}}^*)^m$,\n    the restriction of the parametrized zero set $\\rowv{x}$, defined as\n    \\[\n        \\rowv{x}(\\lambda) = \n        \\left(\n            \\frac{ c e^{\\ensuremath{\\mathfrak{i}} \\lambda} }{2\\ensuremath{\\mathfrak{i}}} \\cdot\n            \\rowv{\\eta}_{\\dig{T}} \\circ\n            \\rowv{a}(\\dig{T})^{-I}\n        \\right)^{\\ensuremath{\\check{Q}}(\\dig{T})^{-1}}\n        \\in (S^1)^m\n    \\]\n    for any $\\lambda \\in \\ensuremath{\\mathbb{R}}$.\n    By \\Cref{thm: unicycle +dimensional} $\\rowv{x}(\\lambda)$ also satisfies the algebraic Kuramoto equations.\n    Therefore, \n    $\\rowv{\\theta}(\\lambda) = \\log(\\rowv{x}(\\lambda))$\n    is a positive-dimensional real zero set of the  transcendental Kuramoto system.\n\\end{proof}\n\\begin{example}[4-cycle, again]\\label{ex: C4 +dimensional}\n    For $G = C_4$, as noted in \\Cref{ex: C4},\n    there can be as many as 6 balanced subnetworks,\n    depending on the choices of coupling coefficients.\n    They come in 3 transpose-pairs\n    whose representatives are shown in \\Cref{fig: C4 classes of facet subgraphs}.\n    Each transpose-pair of balanced subnetworks\n    produces an one-dimensional $\\ensuremath{\\mathbb{C}}^*$-zero sets of $\\colv{f}_G = 0$\n    through the formula given in \\Cref{thm: unicycle +dimensional}.\n    \n    \\begin{enumerate}[topsep=0pt,itemsep=0.75ex,partopsep=0ex,parsep=0ex]\n    \\item \n    Consider the first subdigraph $\\dig{H}_1$ in \\Cref{fig: C4 classes of facet subgraphs}.\n    It is balanced if $k_{10} k_{32} = k_{12} k_{30}$.\n    Then with the choice of $\\dig{T}_1$ having arcs $(1,0), (1,2), (3,0)$,\n    we can compute\n    \\begin{align*}\n        \\ensuremath{\\check{Q}}(\\dig{T}_1) &=\n        \\left[\n        \\begin{smallmatrix}\n            +1 & +1 &  0 \\\\\n             0 & -1 &  0 \\\\\n             0 &  0 & +1\n        \\end{smallmatrix}\n        \\right]\n        &\n        \\ensuremath{\\check{Q}}(\\dig{T}_1)^{-1} &=\n        \\left[\n        \\begin{smallmatrix}\n            +1 & +1 &  0 \\\\\n             0 & -1 &  0 \\\\\n             0 &  0 & +1\n        \\end{smallmatrix}\n        \\right]\n        &\n        \\rowv{\\eta}_{\\dig{T}_1} &=\n        \\begin{bmatrix}\n            -1 & +1 & +1\n        \\end{bmatrix}\n        ,\n    \\end{align*}\n    where the notation is as in \\Cref{thm: unicycle +dimensional}.\n    Then the function $\\rowv{x} : \\ensuremath{\\mathbb{C}}^* \\to (\\ensuremath{\\mathbb{C}}^*)^3$, given by\n    \\begin{align*}\n        \\rowv{x}(\\lambda) &=\n        (\n            \\lambda \\cdot \\rowv{\\eta}_{\\dig{T}_1} \\circ \n            \\begin{bmatrix}\n                a_{10} & a_{12} & a_{30}\n            \\end{bmatrix}^{-I}\n        )^{ \\ensuremath{\\check{Q}}(\\dig{T}_1)^{-1} }\n        =\n        \\begin{bmatrix}\n            -\\frac{2\\ensuremath{\\mathfrak{i}} \\lambda}{k_{10}} & \n            -\\frac{k_{12}}{k_{10}} &\n            \\frac{2\\ensuremath{\\mathfrak{i}} \\lambda}{k_{30}}\n        \\end{bmatrix}\n    \\end{align*}\n    has image inside the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_{C_4}$.\n    In other words, the function\n    \\[\n        \\rowv{x}(\\lambda)\n        =\n        \\begin{bmatrix}\n             -2\\ensuremath{\\mathfrak{i}} \\lambda \/ k_{10} &\n             -k_{12} \/ k_{10} &\n             +2\\ensuremath{\\mathfrak{i}} \\lambda \/ k_{30}\n        \\end{bmatrix}\n        \\quad\\quad\\text{if}\\quad\n        k_{10} k_{32} = k_{12} k_{30}\n    \\]\n    parametrizes the open part of an one-dimensional orbit\n    in the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_{C_4}$.\n    In the special case when $k_{ij} = 1$,\n    this one-dimensional orbit is \\emph{balanced}\n    in the sense of \\cite[Definition 3.1]{sclosa2022kuramoto}\n    and \\cite{sclosa2022kuramoto} provides analysis of the stability and geometry of such orbits.\n    \n    \\item\n    The second subnetwork $\\dig{H}_2$\n    in \\Cref{fig: C4 classes of facet subgraphs}\n    is balanced if $k_{10} k_{30} = k_{12} k_{23}$.\n    With $\\dig{T}_2 < \\dig{H}_2$ given by arcs\n    $(1,0), (1,2), (0,3)$ and  by following the construction above,\n    we have \n    \\[\n        \\rowv{x}(\\lambda)\n        =\n        \\begin{bmatrix}\n             + 2\\ensuremath{\\mathfrak{i}} \\lambda \/ k_{10} &\n             - k_{12} \/ k_{10} &\n             + k_{03} \/ 2\\ensuremath{\\mathfrak{i}} \\lambda\n        \\end{bmatrix}\n        \\quad\\quad\\text{if}\\quad\n        k_{10} k_{30} = k_{12} k_{23}\n    \\]\n    whose image is in the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_{C_4}$\n    for any $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$.\n    Note that even in the special case of $k_{ij} = 1$,\n    this orbit is \\emph{not} balanced\n    (in the sense of \\cite[Definition 3.1]{sclosa2022kuramoto})\n    for $\\lambda \\ne 1$.\n    \n    \\item\n    Finally, the third subnetwork $\\dig{H}_3$\n    in \\Cref{fig: C4 classes of facet subgraphs}\n    is balanced if $k_{10} k_{12} = k_{32} k_{03}$.\n    With the choice $\\dig{T}_3 < \\dig{H}_3$,\n    and following the construction above,\n    we have \n    \\[\n        \\rowv{x}(\\lambda)\n        =\n        \\begin{bmatrix}\n             + k_{01} \/ 2\\ensuremath{\\mathfrak{i}} \\lambda &\n             - k_{01} k_{12} \/ 4 \\lambda^2 &\n             - k_{03} \/ 2\\ensuremath{\\mathfrak{i}} \\lambda\n        \\end{bmatrix}\n        \\quad\\text{if}\\quad\n        k_{10} k_{12} = k_{32} k_{30}\n    \\]\n    whose image is in the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_{C_4}$\n    for any $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$.\n    This orbit is more interesting as all three complex phase variables\n    $x_1,x_2,x_3$ are nonconstant relative to the reference phase $x_0 = 1$.\n    It is also \\emph{not} balanced (in the sense of \\cite[Definition 3.1]{sclosa2022kuramoto}).\n    \\end{enumerate}\n    \n    Moreover, if the coupling coefficients satisfy the condition\n    $\\rowv{k}^2(C_4) = c^2 \\cdot \\rowv{1}$ for some $c \\in \\ensuremath{\\mathbb{R}}$,\n    then the three $\\ensuremath{\\mathbb{C}}^*$-orbits contain one-real-dimensional components\n    on the real torus $(S^1)^3$.\n    Indeed, by the restriction $\\lambda = \\frac{c e^{\\ensuremath{\\mathfrak{i}} t}}{2 \\ensuremath{\\mathfrak{i}}}$,\n    the three orbits constructed above reduce to parametrized zero sets\n    of one real-dimension inside the real torus.\n    Here, $\\sgn(x) \\in \\{ \\pm 1 \\}$ denotes the sign of $x \\in \\ensuremath{\\mathbb{R}}$.\n    If we define\n    \\[\n        \\sigma^{ij}_{\\ell m} =\n        \\begin{cases}\n            0 &\\text{if } k_{ij}\/k_{\\ell m} > 0 \\\\\n            \\pi &\\text{otherwise},\n        \\end{cases}\n        \\quad\\text{and}\\quad\n        \\sigma^{ij} =\n        \\begin{cases}\n            0 &\\text{if } k_{ij} > 0 \\\\\n            \\pi &\\text{otherwise},\n        \\end{cases}\n    \\]\n    then the three potential real orbits can be expressed as\n    {\\small\n    \\begin{align}\n        &\n        \\left\\{\n        \\begin{aligned}\n            \\theta_1 &= t + \\pi + \\sigma^{10} \\\\\n            \\theta_2 &= \\pi + \\sigma^{12}_{10} \\\\\n            \\theta_3 &= t + \\sigma^{30} \\\\\n        \\end{aligned}\n        \\right.\n        \\text{if } \\frac{k_{10} k_{32}}{k_{12} k_{30}} = 1,\n        &&\n        \\left\\{\n        \\begin{aligned}\n            \\theta_1 &= \\sigma^{10} + t \\\\\n            \\theta_2 &= \\pi + \\sigma^{21}_{10} \\\\\n            \\theta_3 &= \\sigma^{30} - t\n        \\end{aligned}\n        \\right.\n        \\text{if } \n        \\frac{k_{10} k_{30}}{ k_{12} k_{23} } = 1,\n        &&\n        \\left\\{\n        \\begin{aligned}\n            \\theta_1 &= \\sigma^{10} - t \\\\\n            \\theta_2 &= \\sigma^{21}_{10} -2t \\\\ \n            \\theta_3 &= \\pi + \\sigma^{30} -t \\\\\n        \\end{aligned}\n        \\right.\n        \\text{if }\n        \\frac{ k_{10} k_{12} }{ k_{32} k_{30} } = 1,\n        \\label{equ: C4 components}\n    \\end{align}\n    }%\n    with $\\theta_0 = 0$.\n    It is easy to see if $k_{ij}$'s are identical,\n    then all three real orbits exist.\n    Moreover, these three orbits intersects at a singular point\n    $(\\theta_0,\\theta_1,\\theta_2,\\theta_3) = (0, \\pi\/2, \\pi, \\pi\/2)$.\n    These positive-dimensional solution sets have been studied in Refs.~\\cite{LindbergZachariahBostonLesieutre2022Distribution,sclosa2022kuramoto}\n    (\\cite[Section 5.2]{sclosa2022kuramoto}, in particular,\n    also provide topological analysis for the orbits).\n    In this example, we showed they can also be derived systematically\n    from balanced subnetworks.\n\\end{example}\n\n\\subsection{Multiple even cycles sharing one edge}\n\nWe now show \npositive-dimensional\nzero sets for $\\colv{f}_G$ \nis possible for networks consisting of\nmultiple even cycles sharing a single edge\n(e.g., \\Cref{fig: 8 again}).\n\n\\begin{proposition}\n    Suppose $G$ consists of $d$ independent even cycles\n    $C_1,\\ldots,C_d$ that share a single edge $e$,\n    then with the choice of the coupling coefficients\n    \\[\n        k_{ij} =\n        \\begin{cases}\n            sd &\\text{if } \\{i,j\\} = e \\\\\n            s  &\\text{othewise},\n        \\end{cases}\n    \\]\n    for any $s \\in \\ensuremath{\\mathbb{C}}^*$,\n    and homogeneous natural frequencies $\\colv{w} = w \\cdot \\colv{1}$\n    for any $w \\in \\ensuremath{\\mathbb{C}}^*$,\n    the algebraic Kuramoto system $\\colv{f}_G$\n    derived from the network $(G,K,w \\cdot \\colv{1})$\n    has a positive-dimensional $\\ensuremath{\\mathbb{C}}^*$-zero set.\n\\end{proposition}\n\\begin{proof}\n    Since $G$ is bipartite, by \\Cref{thm: faces are max bipartite},\n    there exists a facet $F$ of $\\ensuremath{\\check{\\nabla}}_G$ such that $G_F = G$,\n    and thus $\\ensuremath{\\check{\\nabla}}_G = \\{ \\colv{0} \\} \\cup F \\cup (-F)$.\n    Consequently,\n    \\[\n        \\colv{f}_G(\\rowv{x}) =\n        \\ensuremath{\\check{Q}}(\\dig{G}_F   )( \\rowv{x}(\\lambda)^{\\ensuremath{\\check{Q}}(\\dig{G}_F   )} \\circ \\rowv{a}(\\dig{G}_F)    )^\\top +\n        \\ensuremath{\\check{Q}}(\\dig{G}_{-F})( \\rowv{x}(\\lambda)^{\\ensuremath{\\check{Q}}(\\dig{G}_{-F})} \\circ \\rowv{a}(\\dig{G}_{-F}) )^\\top.\n    \\]\n    Let $T$ be a spanning tree of $G$ that contains $e$,\n    and let $\\dig{T}$ be the corresponding subdigraph such that\n    $\\dig{T} < \\dig{G}_F$.\n    We shall arrange of the columns of $\\ensuremath{\\check{Q}}(\\dig{G}_F)$ and $\\ensuremath{\\check{Q}}(\\dig{T})$\n    so that\n    \\begin{align*}\n        \\ensuremath{\\check{Q}}(\\dig{G}_F) &=\n        \\begin{bmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{T}) & \\colv{v}_1 & \\cdots & \\colv{v}_d\n        \\end{bmatrix}\n        &&\\text{and} &\n        \\ensuremath{\\check{Q}}(\\dig{T}) &=\n        \\begin{bmatrix}\n            \\ensuremath{\\check{Q}}(\\dig{T} - \\{ e \\}) & \\colv{v}_e\n        \\end{bmatrix}\n    \\end{align*}\n    where $\\colv{v}_e$ is the incidence vector corresponding to the edge $e$\n    that is shared by all cycles,\n    and $\\colv{v}_1,\\ldots,\\colv{v}_d$ are the incidence vectors corresponding to\n    the arcs associated with the edges $e_1,\\ldots,e_d$ in\n    $\\ensuremath{\\mathcal{E}}(C_1) \\setminus \\ensuremath{\\mathcal{E}}(T), \\ldots, \\ensuremath{\\mathcal{E}}(C_d) \\setminus \\ensuremath{\\mathcal{E}}(T)$, respectively.\n    Pick basis vectors $\\colv{\\eta}_1,\\ldots,\\colv{\\eta}_d$ of $\\ker \\ensuremath{\\check{Q}}(\\dig{G}_F)$\n    that are primitive cycle vector associated with the cycles $C_1,\\ldots,C_d$\n    with consistent signs\n    (i.e., their shared nonzero entries have the same sign).\n    Let\n    \\[\n        \\rowv{\\eta} = \n        \\begin{bmatrix}\n            \\rowv{\\eta}_T & u_1 & \\cdots & u_d\n        \\end{bmatrix}\n        :=\n        (\\colv{\\eta}_1 + \\cdots + \\colv{\\eta}_d)^\\top.\n    \\]\n    Then the first $n-1$ entries of $\\rowv{\\eta}_T$ are $\\pm 1$,\n    and its $n$-th entry is $\\pm d$.\n    The signs depend on the choices of the primitive cycle vectors.\n    \n    With these, we define the nonconstant function\n    $\\rowv{x} : \\ensuremath{\\mathbb{C}}^* \\to (\\ensuremath{\\mathbb{C}}^*)^n$, given by\n    \\[\n        \\rowv{x}(\\lambda) = (\n            \\lambda \\cdot\n            \\rowv{\\eta}_{\\dig{T}} \\circ\n            \\rowv{a}(\\dig{T})^{-I}\n        )^{\\ensuremath{\\check{Q}}(\\dig{T})^{-1}},\n    \\]\n    as above. Then, as demonstrated in \\Cref{thm: facial cycle condition},\n    \\[\n        \\ensuremath{\\check{Q}}(\\dig{G}_F)( \\rowv{x}(\\lambda)^{\\ensuremath{\\check{Q}}(\\dig{G}_F)} \\circ \\rowv{a}(\\dig{G}_F) )^\\top = \\colv{0}\n    \\]\n    for any $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$.\n    It remains to show \n    $\\ensuremath{\\check{Q}}(\\dig{G}_{-F})( \\rowv{x}(\\lambda)^{\\ensuremath{\\check{Q}}(\\dig{G}_{-F})} \\circ \\rowv{a}(\\dig{G}_{-F}) )^\\top$\n    also vanishes.\n    Since $\\rowv{a}(\\dig{T}) = \\frac{\\rowv{k}(\\dig T)}{2\\ensuremath{\\mathfrak{i}}} = [\\;s \\; \\cdots \\; s \\; sd\\;]$,\n    \\begin{align*}\n        \\rowv{x}(\\lambda)^{ - \\ensuremath{\\check{Q}}(\\dig T) } \\circ \\rowv{a}(\\dig T)\n        &=\n        (\n            \\lambda \\cdot\n            \\rowv{\\eta}_{\\dig{T}} \\circ\n            \\rowv{a}(\\dig{T})^{-I}\n        )^{-I}\n        \\circ \\rowv{a}(\\dig T)\n        = \n        \\lambda^{-1} \n        \\rowv{\\eta}_{\\dig{T}}^{-I} \\circ\n        \\begin{bmatrix}\n            \\frac{s^2}{-4} & \\cdots & \\frac{s^2}{-4} & \\frac{s^2 d^2}{-4}\n        \\end{bmatrix}\n        = \n        \\frac{s^2}{-4\\lambda} \\,\n        \\rowv{\\eta}_{\\dig{T}}\n    \\end{align*}\n    Moreover, by construction,\n    $\\rowv{x}(\\lambda)^{ \\colv{v}_i } \\cdot \\rowv{a}(e_i) = \\lambda u_i$ \n    for $i=1,\\ldots,d$.\n    Therefore,\n    \\begin{align*}\n        \\rowv{x}(\\lambda)^{ - \\colv{v}_i } \\cdot \\rowv{a}(e_i)\n        &= \n        \\lambda^{-1} \\rowv{a}^2(e_i) u_i^{-1} \n        =\n        \\frac{s^2}{-4 \\lambda} \\, u_i,\n    \\end{align*}\n    since $u_1 \\in \\{ \\pm 1 \\}$.\n    This shows that\n    \\begin{align*}\n        \\rowv{x}(\\lambda)^{ \\ensuremath{\\check{Q}}(\\dig{G}_{-F}) } \\circ \\rowv{a}(\\dig{G}_{-F})\n        &=\n        \\begin{bmatrix}\n            \\rowv{x}(\\lambda)^{-\\ensuremath{\\check{Q}}(\\dig T) } \\circ \\rowv{a}(\\dig T) &\n            \\rowv{x}(\\lambda)^{ - \\colv{v}_1 } \\cdot \\rowv{a}(e_1) &\n            \\rowv{x}(\\lambda)^{ - \\colv{v}_2 } \\cdot \\rowv{a}(e_1)\n        \\end{bmatrix}\n        =\n        \\frac{s^2}{-4 \\lambda} \\, \\rowv{\\eta}.\n    \\end{align*}\n    Consequently, $\\colv{f}_G(\\rowv{x}(\\lambda)) = \\colv{0}$ for any $\\lambda \\in \\ensuremath{\\mathbb{C}}^*$,\n    i.e., the $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_G$ is positive-dimensional.\n\\end{proof}\n\n\\begin{figure}[h]\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[\n            scale=1.5,\n            main\/.style = {draw,circle,minimum size=1.5ex},\n            every edge\/.style = {draw,thick}\n        ] \n            \\node[main] (0) at ( 0, 0) {$0$}; \n            \\node[main] (1) at ( 1, 0) {$1$}; \n            \\node[main] (2) at ( 2, 0) {$2$}; \n            \\node[main] (3) at ( 2, 1) {$3$}; \n            \\node[main] (4) at ( 1, 1) {$4$}; \n            \\node[main] (5) at ( 0, 1) {$5$}; \n            \\path (0) edge node[below] {$s$} (1);\n            \\path (1) edge node[below] {$s$} (2);\n            \\path (2) edge node[right] {$s$} (3);\n            \\path (3) edge node[above] {$s$} (4);\n            \\path (4) edge node[above] {$s$} (5);\n            \\path (5) edge node[left]  {$s$} (0);\n            \\path (1) edge node[right] {$2s$} (4);\n        \\end{tikzpicture} \n        \\caption{$G$}\n        \\label{fig: 8 again}\n    \\end{subfigure}~%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[\n            scale=1.5,\n            main\/.style = {draw,circle,minimum size=1.5ex},\n            every edge\/.style = {draw,thick,-latex}\n        ] \n            \\node[main] (0) at ( 0, 0) {$0$}; \n            \\node[main] (1) at ( 1, 0) {$1$}; \n            \\node[main] (2) at ( 2, 0) {$2$}; \n            \\node[main] (3) at ( 2, 1) {$3$}; \n            \\node[main] (4) at ( 1, 1) {$4$}; \n            \\node[main] (5) at ( 0, 1) {$5$}; \n            \\path (1) edge (0);\n            \\path (1) edge (2);\n            \\path (3) edge (2);\n            \\path (3) edge (4);\n            \\path (5) edge (4);\n            \\path (5) edge (0);\n            \\path (1) edge (4);\n        \\end{tikzpicture} \n        \\caption{A facet subdigraph $\\dig{G}_F$}\n        \\label{fig: 8 facet again}\n    \\end{subfigure}~%\n    \\begin{subfigure}{0.3\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[\n            scale=1.5,\n            main\/.style = {draw,circle,minimum size=1.5ex},\n            every edge\/.style = {draw,thick,-latex}\n        ] \n            \\node[main] (0) at ( 0, 0) {$0$}; \n            \\node[main] (1) at ( 1, 0) {$1$}; \n            \\node[main] (2) at ( 2, 0) {$2$}; \n            \\node[main] (3) at ( 2, 1) {$3$}; \n            \\node[main] (4) at ( 1, 1) {$4$}; \n            \\node[main] (5) at ( 0, 1) {$5$}; \n            \\path (1) edge (0);\n            \\path (1) edge (2);\n            \\path (3) edge (2);\n            \\path (5) edge (0);\n            \\path (1) edge (4);\n        \\end{tikzpicture} \n        \\caption{$\\dig{T}$ inside $\\dig{G}_F$}\n        \\label{fig: 8 T}\n    \\end{subfigure}\n    \\caption{\n        A network with two independent even cycles.\n    }\n\\end{figure}\n\\begin{example}\n    Consider the network shown in \\Cref{fig: 8 again}.\n    We fix a particular facet\n    whose facet subdigraph is shown in \\Cref{fig: 8 facet again}.\n    We also fix a choice of $\\dig{T} < \\dig{G}_F$ shown in \\Cref{fig: 8 T}.\n    With this choice of $\\dig{T}$ and the ordering of the arcs\n    $(1,0)$, $(1,2)$, $(3,2)$, $(5,0)$, $(1,4)$, $(3,4)$, $(5,4)$,\n    we have\n    \\begin{align*}\n        \\ensuremath{\\check{Q}}(\\dig T) &=\n        \\left[\n        \\begin{smallmatrix}\n             1 &  1 & 0 & 0 & 1 \\\\\n             0 & -1 &-1 & 0 & 0 \\\\\n             0 &  0 & 1 & 0 & 0 \\\\\n             0 &  0 & 0 & 0 &-1 \\\\\n             0 &  0 & 0 & 1 & 0   \n        \\end{smallmatrix}\n        \\right]\n        ,\n        &\n        \\ensuremath{\\check{Q}}(\\dig T)^{-1} &=\n        \\left[\n        \\begin{smallmatrix}\n             1 &  1 & 1 & 1 & 0 \\\\\n             0 & -1 &-1 & 0 & 0 \\\\\n             0 &  0 & 1 & 0 & 0 \\\\\n             0 &  0 & 0 & 0 & 1 \\\\\n             0 &  0 & 0 &-1 & 0   \n        \\end{smallmatrix}\n        \\right]\n        ,\n        &\n        \\colv{\\eta}_1 &=\n        \\left[\n        \\begin{smallmatrix}\n             0 \\\\\n            -1 \\\\\n            +1 \\\\\n             0 \\\\\n            +1 \\\\\n            -1 \\\\\n             0\n        \\end{smallmatrix}\n        \\right]\n        ,\n        &\n        \\colv{\\eta}_2 &=\n        \\left[\n        \\begin{smallmatrix}\n            -1 \\\\\n             0 \\\\\n             0 \\\\\n            +1 \\\\\n            +1 \\\\\n             0 \\\\\n            -1\n        \\end{smallmatrix}\n        \\right]\n    \\end{align*}\n    as constructed in the proof above.\n    We also define\n    \\begin{align*}\n        \\rowv{\\eta} &= \\colv{\\eta}_1^\\top + \\colv{\\eta}_2^\\top =\n        \\begin{bmatrix}\n            -1 &\n            -1 &\n            +1 &\n            +1 &\n            +2 &\n            -1 &\n            -1\n        \\end{bmatrix}\n        &&\\text{and} &\n        \\rowv{\\eta}_T &=\n        \\begin{bmatrix}\n            -1 &\n            -1 &\n            +1 &\n            +1 &\n            +2\n        \\end{bmatrix}.\n    \\end{align*}\n    We then verify that the construction\n    $\n        \\rowv{x}(\\lambda) = \n        (\n            \\lambda \\cdot\n            \\rowv{\\eta}_T\n            \\circ\n            \\rowv{a}(\\dig T)^{-I}\n        )^{ \\ensuremath{\\check{Q}}(\\dig T)^{-1} }\n    $\n    above produces\n    \\begin{align*}\n        &\n        (\n            \\lambda\n            \\begin{bmatrix}\n                -1 &\n                -1 &\n                +1 &\n                +1 &\n                +2\n            \\end{bmatrix}\n            \\circ\n            \\begin{bmatrix}\n                \\frac{s}{2\\ensuremath{\\mathfrak{i}}} & \\frac{s}{2\\ensuremath{\\mathfrak{i}}} & \\frac{s}{2\\ensuremath{\\mathfrak{i}}} & \\frac{s}{2\\ensuremath{\\mathfrak{i}}} & \\frac{2s}{2\\ensuremath{\\mathfrak{i}}}\n            \\end{bmatrix}^{-I}\n        )^{\n            \\ensuremath{\\check{Q}}(\\dig T)^{-1}\n        }\n        &=\n        \\begin{bmatrix}\n            - \\frac{2\\ensuremath{\\mathfrak{i}} \\lambda}{s} &\n            +1 & \n            + \\frac{2\\ensuremath{\\mathfrak{i}} \\lambda}{s} &\n            -1 &\n            + \\frac{2\\ensuremath{\\mathfrak{i}} \\lambda}{s}\n        \\end{bmatrix}\n        ,\n    \\end{align*}\n    which\n    parametrizes an one-dimensional $\\ensuremath{\\mathbb{C}}^*$-zero set of $\\colv{f}_G = 0$.\n    Moreover, by choosing $\\lambda(t) = \\frac{s e^{\\ensuremath{\\mathfrak{i}} t}}{2 \\ensuremath{\\mathfrak{i}}}$,\n    $\\rowv{x} \\in (S^1)^5$ for any $t \\in \\ensuremath{\\mathbb{R}}$.\n    Therefore, $\\log(\\rowv{x})$ produces the one-dimensional real zero set\n    \\[\n        (\\theta_0,\\theta_1,\\theta_2,\\theta_3,\\theta_4,\\theta_5) =\n        (\n            0, \n            t + \\pi,\n            0,\n            t,\n            \\pi,\n            t\n        )\n    \\]\n    for the transcendental Kuramoto system \\eqref{equ: kuramoto sin}\n    derived from the network in \\Cref{fig: 8 again}.\n\\end{example}\n\n\\section{Concluding remarks}\\label{sec: conclusion}\n\nWe studied the structure of the zero sets of the\nKuramoto equations and its algebraic counterpart.\nBy leveraging a recently discovered\nthree-way connection between the\ngraph-theoretic, convex-geometric, and tropical view points,\nwe answered four key questions.\n\n\\begin{enumerate}\n    \\item\n    For \\Cref{q1},\n    we showed that\n    \\emph{%\n        for generic natural frequencies and\n        generic but symmetric coupling coefficients,\n        the $\\ensuremath{\\mathbb{C}}^*$ root count for the algebraic Kuramoto system\n        coincides with the adjacency polytope bound,\n        and as a corollary we showed that the algebraic Kuramoto system is Bernshtein-general.\n    }\n    \n    \\item\n    For \\Cref{q2}, we provided a description of the\n    \\emph{\n        exceptional coupling coefficients\n        for which the $\\ensuremath{\\mathbb{C}}^*$ root count of the algebraic Kuramoto system\n     drops below the generic zero count.\n    }\n        This description used graph-theoretic, combinatorial and toric information.\n    \n    \\item \n    For \\Cref{q3}, we analyzed unicycle networks and\n    developed a\n    \\emph{\n        full stratification of the coupling coefficient space\n        and computed the $\\ensuremath{\\mathbb{C}}^*$ root count over each stratum.\n    }\n    \n    \\item \n    For \\Cref{q4},\n    we established sufficient conditions\n    on unicycle networks and networks consisting of cycles sharing a single edge\n    under which \n    \\emph{\n        there will be non-isolated real and complex\n        zero sets for Kuramoto systems\n    }\n    through explicit constructions.\n\\end{enumerate}\n\nWhile the analysis for the last two questions required some\ntopological restrictions,\nit appears hopeful that\nthe approach taken here can be generalized to other networks.\nWe hope our work will spark interest in the\nfull analysis of the typical and atypical solutions to Kuramoto equations.\n\nThe proofs are constructive.\nIn particular, the homotopies used in\n\\Cref{thm: generic root count} and \\Cref{lem: pyramid system}\nare specialized polyhedral homotopy \\cite{HuberSturmfels1995Polyhedral}\nthat will likely bring significant performance improvement.\n\nFinally, we speculate that the explicit connection between the root count\nof the algebraic Kuramoto equations and the normalized volume of adjacency polytopes may also allow algebraic geometers\nto directly contribute to the geometric study of adjacency polytopes.\n\n\\section*{Acknowledgments}\n\nThis project is inspired by a series of discussion\nthe first named author had with\nAnton Leykin, Josephine Yu, and Yue Ren\nbetween 2017 and 2018.\nThe first named author also learned much about\nthe structure of adjacency polytopes (a.k.a. symmetric edge polytopes)\nfrom Robert Davis,\nAlessio D'Al\\`i, Emanuele Delucchi, and Mateusz Micha\\l{}ek.\nThe authors thank Paul Breiding, Paul Helminck, and Davide Sclosa\nfor their comments on an earlier version of this manuscript.\n\n\n\n\n\\section{Elementary lemmata}\\label{sec: lemmas}\n\nWe restate \\Cref{lem: simplex tree}\nand provide an elementary proof.\n\n\\simplextree*\n\n\\begin{proof}\n    Let $(\\colv{\\alpha},1)$ be the upward pointing inner normal\n    that defines the cell $\\Delta$ as a projection of a lower facet of $\\ensuremath{\\check{\\nabla}}_G^{\\omega}$.\n    Suppose $\\dig{G}_\\Delta$ has a simple directed cycle $i_1 \\to \\cdots \\to i_m \\to i_1$.\n    Then\n    \\[\n        \\inner{ \\colv{\\alpha} }{ \\colv{e}_{i_k} - \\colv{e}_{i_{k+1}} } + 1 + \\delta_{i_k,i_{k+1}} = 0\n        \\quad\\text{for } k = 1,\\ldots,m,\n    \\]\n    where $i_{m+1} = i_1$.\n    Summing these $m$ equations produces\n    $\n        m + \\sum_{k=1}^m \\delta_{i_k,i_{k+1}} = 0,\n    $\n    which is not possible under the assumption that $\\delta_{ij}$\n    are sufficiently close to 0.\n    So $\\dig{G}_\\Delta$ must be acyclic.\n    \n    Moreover, $\\dim(\\Delta) = n$, by assumption.\n    That is, $\\{ \\colv{e}_i - \\colv{e}_j \\mid (i,j) \\in \\dig{G}_\\Delta \\}$\n    is must span $\\ensuremath{\\mathbb{R}}^n$ as a set of vectors\n    (since $\\colv{0} \\in \\Delta$, by construction).\n    Therefore, for every $i \\in \\{0,\\ldots,n\\}$,\n    either $\\colv{e}_i - \\colv{e}_j \\in \\Delta$\n    or     $\\colv{e}_j - \\colv{e}_i \\in \\Delta$\n    for some $j \\in \\{0,\\ldots,n\\} \\setminus \\{i\\}$.\n    That is, $G_\\Delta$ must be spanning.\n\\end{proof}\n\n\\begin{lemma}\nRecall the set up in \\Cref{ex: C4}. For each stratification of the coupling coefficients, there exist parameter values where all $\\mathbb{C}^*$ solutions are real.\n\\end{lemma}\n\\begin{proof}\nRecall the stratification of the coupling coefficients for $C_4$ given in \\Cref{ex: C4}. By \\Cref{pro: unicycle balanced count}, we have an upper bound on the number of $\\ensuremath{\\mathbb{C}}^*$ (and therefore $\\ensuremath{\\mathbb{R}}^*$) solutions in each case. Using the parameter values given in \\Cref{ex: C4}, we use \\texttt{HomotopyContinuation.jl} to find all complex solutions and certify they are real using interval arithmetic \\cite{homotopyjl,breiding2020certifying}. Since the upper bound in \\Cref{pro: unicycle balanced count} matches the number of certified complex solutions, the result follows. \n\\end{proof}\n\n\\section{Regular zeros and the principle of homotopy continuation}\\label{sec: homotopy}\n\nAn isolated zero $\\rowv{x} \\in (\\ensuremath{\\mathbb{C}}^*)^n$ of a square Laurent system $\\colv{f}$\nis said to be \\emph{regular}\nif the Jacobian matrix $D \\colv{f}$ is invertible at $\\rowv{x}$.\nOtherwise, it is \\emph{singular}.\nThis definition applies to\nboth the interpretation $\\colv{f} : (\\ensuremath{\\mathbb{C}}^*)^n \\to (\\ensuremath{\\mathbb{C}}^*)^n$, as a holomorphic function\nand  the interpretation $\\colv{f} : ((\\ensuremath{\\mathbb{R}}^*)^2 \\setminus \\{0\\})^n \\to \\ensuremath{\\mathbb{R}}^{2n}$,\nas a smooth function between real manifolds.\nSard's Theorem\nstates that for almost all choices of $\\colv{\\epsilon}$,\nall zeros of the square system $\\colv{f}(\\rowv{x}) - \\colv{\\epsilon}$ are regular.\nIn other words, having regular zeros is a ``generic'' behavior.\n\nIn the arguments presented in this paper,\nwe made frequent reference of a collection of theorems\nfrom differential geometry that we simply referred to as\nthe \\emph{principle of homotopy continuation}.\nThis is a deep theoretical framework that has found\nmany important applications.\nHere, we give an overly simplified description.\nConsider a smooth function $H : M \\times \\ensuremath{\\mathbb{R}} \\to N$\nwhere $M$ and $N$ are manifolds of the same dimension.\nSuppose $\\rowv{x}_0 \\in M$ is a regular zero of $H(\\cdot,t_0)$\nfor some $t_0 \\in \\ensuremath{\\mathbb{R}}$,\nthen there exists a smooth function $\\rowv{x}(t) : [t_0,t_1) \\to M$,\nfor some $t_1 > t_0$,\nsuch that $\\rowv{x}(t_0) = \\rowv{x}_0$ and $H(\\rowv{x}(t),t) = \\colv{0}$.\nThere are exactly three possibilities for the domain $[t_0,t_1)$ of $\\rowv{x}(t)$:\n(A) $t_1 = \\infty$;\n(B) $\\rowv{x}(t_1)$ is a singular zero of $H(\\cdot,t_0)$; or\n(C) $\\rowv{x}(t)$ converges to a limit point outside $M$ as $t \\to t_1$.\nThis can be established through repeated applications of\nInverse Function Theorem and Implicit Function Theorem.\nIts complex version can be established as a special case of analytic continuations.\n\n\\section{Notations}\\label{app: notations}\nHere, we list notations used in this paper\nthat may not be standard.\n\n\\begin{description}[itemsep=1.2ex]\n\\item[$\\ensuremath{\\mathbb{C}}\\{\\tau\\}$]\n    The field of Puiseux series in $\\tau$ with complex coefficients\n    is denoted $\\ensuremath{\\mathbb{C}}\\{\\tau\\}$,\n    and only convergent series,\n    representing germs of one-dimensional analytic varieties,\n    will be relevant.\n    \n\\item[$\\mathcal{C}(P,\\Delta_\\omega)$]\n    The (closed) secondary cone of a regular subdivision $\\Delta_\\omega$\n    in a polytope $P$.\n    \n\\item[$\\Sigma_{\\omega}(P)$]\n    The regular subdivision of a point configuration $P$\n    induced by a lifting function $\\omega : P \\to \\mathbb{Q}$.\n    \n\\item[$\\ensuremath{\\check{\\nabla}}_G$]\n    The point configuration associated with the adjacency polytope\n    derived from a connected graph $G$.\n    It is defined to be\n    $\\{ \\pm ( \\colv{e}_i - \\colv{e}_j ) \\mid \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G) \\} \\cup \\{ \\colv{0} \\}$.\n\n\\item[$\\ensuremath{\\check{\\nabla}}_G^\\omega$]\n    ``Lifted'' version of point configuration $\\ensuremath{\\check{\\nabla}}_G$\n    induced by a lifting function $\\omega : \\ensuremath{\\check{\\nabla}}_G \\to \\mathbb{Q}$.\n    It consists of the points $(\\colv{p}, \\omega(\\colv{p})) \\subset \\ensuremath{\\mathbb{R}}^{n+1}$\n    for all $\\colv{p} \\in \\ensuremath{\\check{\\nabla}}_G$.\n    \n\\item[$\\colv{f}_{(G,K,\\colv{w})}, \\colv{f}_{(G,K)}, \\colv{f}_{G}$]\n    The algebraic Kuramoto system derived from a Kuramoto network $(G,K,\\colv{w})$.\n    If the choice of the natural frequencies $\\colv{w}$ is not relevant to the discussions\n    (or assumed to be generic),\n    the notation $\\colv{f}_{(G,K)}$ will be used.\n    Similarly, if only graph topology is of relevance,\n    we will simply use $\\colv{f}_G$.\n    \n\\item[$\\colv{f}^*_G$]\n    The randomized algebraic Kuramoto system,\n    i.e., $R \\cdot \\colv{f}_G$ for a generic square matrix $R$.\n    \n\\item[$(G,K,\\colv{w}), (G,K)$]\n    A Kuramoto network.\n    $G$ is the underlying graph,\n    $K = \\{ k_{ij} \\mid \\{i,j\\} \\in \\ensuremath{\\mathcal{E}}(G) \\}$ with $k_{ij} = k_{ji}$\n    encodes the coupling coefficients,\n    and $\\colv{w} = (w_0,\\ldots,w_n)^\\top$ contains the natural frequencies.\n\n\\item[$\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(f)$]\n    For a Laurent polynomial $f$ in $x_1,\\ldots,x_n$,\n    the \\emph{initial form} of $f$ with respect to a vector $\\rowv{v}$\n    is the Laurent polynomial\n    $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(f)(\\rowv{x}) := \\sum_{\\colv{a} \\in (S)_{\\colv{v}}} c_{\\colv{a}} \\, \\rowv{x}^{\\colv{a}}$, \n    where $(S)_{\\colv{v}}$ is the subset of $S$ on which\n    $\\inner{ \\rowv{v} }{ \\cdot }$ is minimized.\n    It is denoted $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(f)$.\n    For a system $\\colv{f} = (f_1,\\ldots,f_q)$ of Laurent polynomials,\n    $\\ensuremath{\\operatorname{init}}_{\\rowv{v}}(\\colv{f}) = (\\ensuremath{\\operatorname{init}}_{\\rowv{v}} (f_1),\\ldots,\\ensuremath{\\operatorname{init}}_{\\rowv{v}} (f_q))$.\n    \n\\item[$Q(\\dig{H}),\\ensuremath{\\check{Q}}(\\dig{H})$]\nFor a digraph $\\dig{H}$, its \\emph{incidence matrix} $Q(\\dig{H})$\nis the matrix with columns $\\colv{e}_i - \\colv{e}_j$\nsuch that $(i,j) \\in \\ensuremath{\\mathcal{E}}(\\dig{G}_F)$.\nSince we set $\\colv{e}_0 $ to be the zero vector,\nthe first row is all zeros.\nTherefore, we instead consider the \\emph{reduced incidence matrix},\n$\\ensuremath{\\check{Q}}(\\dig H)$, with $n = |\\ensuremath{\\mathcal{V}}(\\dig H)| - 1$ rows,\nwhich is the incidence matrix of $\\dig{H}$ with the first row deleted.\n   \n   \n   \n    The ordering of the columns in both is arbitrary,\n    but, when appears in the same context with other incidence vectors,\n    a consistent ordering is assumed.\n    Here, the adjective ``reduced'' emphasize the fact that\n    the labels for the nodes in the graph are $0,1,\\ldots$,\n    and therefore, for a digraph $\\dig{H}$ of $n+1$ nodes,\n    $\\ensuremath{\\check{Q}}(\\dig{H})$ only has $n$ rows.\n\n\\item[$\\operatorname{MV}(P_1,\\ldots,P_n)$]\n    Given $n$ convex polytopes $P_1,\\ldots,P_n \\subset \\ensuremath{\\mathbb{R}}^n$,\n    the \\emph{mixed volume} of $P_1,\\ldots P_n$\n    is the coefficient of the monomial $\\lambda_1 \\cdots \\lambda_n$\n    in the homogeneous polynomial\n    $\\operatorname{vol}_n(\\lambda_1 P_1 + \\ldots + \\lambda_n P_n)$\n    where $P + Q = \\{ p + q \\ : \\ p \\in P, \\ q \\in Q \\}$\n    denotes the Minkowski sum and $\\operatorname{vol}_n$\n    is the standard $n-$dimensional Euclidean volume form.\n    \n\\item[$\\ensuremath{\\mathcal{N}}_G(i),\\ensuremath{\\mathcal{N}}_{\\dig G}^+(i),\\ensuremath{\\mathcal{N}}_{\\dig G}^-(i)$]\n    For a graph $G$ and a node $i$ of $G$,\n    $\\ensuremath{\\mathcal{N}}_G(i)$ is the set of nodes that are adjacent to $i$.\n    Similarly, the other two are the adjacent nodes\n    through outgoing and incoming arcs in a digraph, respectively.\n    \n\\item[$\\nvol$]\n    The normalized volume of a set in $\\ensuremath{\\mathbb{R}}^n$,\n    which is defined to be $n!$ times the Euclidean volume form.\n    The usage is restricted to convex polytopes here,\n    and we adopt the convention that $\\nvol(X) = 0$,\n    if is not full dimensional.\n\\end{description}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n Frames were first introduced in 1952 by Duffin and Schaeffer \\cite{JA}. In 2000, Frank-Larson \\cite{MD} introduced the notion of frames in Hilbert $ C^*$-modules as a generalization of frames in Hilbert spaces  and Jing \\cite{WJ} continued  to consider them. It is well known that Hilbert $C^*$-modules are  generalizations of Hilbert spaces by allowing the inner product to take values in a $C^*$-algebra rather than in the field of complex numbers. The theory of $2$-inner product spaces as well as an extensive list of related references can be found in \\cite{SG,YPL}. The concept of 2-frames for 2-inner product spaces was introduced by A. Arefijamal and Ghadir Sadeghi \\cite{AS} and described some fundamental properties of them. \nRecently T. Mehdiabad and A. Nazari \\cite{TMN} introduced the $\\mathcal A$-2-inner product space and investigate some inequalities in these spaces. The authors \\cite{MS} defined a 2-inner product that takes values in a locally $C^*$-algebra and studied some properties of it.\\\\   \nIn this paper, we introduce an $\\mathcal A$-2-frame in the $\\mathcal A$-2-inner product space and describe some fundamental properties of them. The tensor product of $\\mathcal A$-2-frames in the $\\mathcal A$-2-inner product space is introduced. It is shown that the tensor product of two $\\mathcal A$-2-frames is an $\\mathcal A$-2-frame for the tensor product of $\\mathcal A$-2-inner product space. Also, we investigate tensor products of $\\mathcal A$-2-frames.\nFrom now,  $\\mathcal A$ denotes a $C^*$-algebra.\n\\begin{definition} \n A pre-Hilbert $\\mathcal A$-module is a complex vector space $E$ which is also a left $\\mathcal A$-module, compatible with the Complex algebra structure,  equipped with an $\\mathcal A$-valued inner product \\\\$\\langle .,.\\rangle:E\\times E\\to \\mathcal A $  which is $\\mathbb C$-linear and $\\mathcal A$-linear in its second variable and satisfies the following relations\\\\\n$(I_1)$  $\\langle x ,x \\rangle\\geq0$  for  every $ x\\in E$,\\\\\n$(I_2)$  $\\langle x ,y \\rangle =\\langle y ,x \\rangle^*$ for  every $ x ,y \\in E$,\\\\\n$(I_3)$  $\\langle x ,x \\rangle =0 $ if  and  only  if $  x=0 $,\\\\\n$(I_4)$  $\\langle ax ,by \\rangle =a^*\\langle x ,y \\rangle b $ for  every $  x ,y \\in E$  and $ a ,b \\in \\mathcal A$,\\\\\n$(I_5)$  $\\langle x ,\\alpha y +\\beta z\\rangle =\\alpha\\langle x ,y\\rangle +\\beta\\langle x ,z\\rangle $  for  every $x ,y ,z \\in E$ and $  \\alpha ,\\beta \\in \\mathbb{C}$.\n\\end{definition}\n\\begin{example}\nLet $l^{2}(\\mathcal A)$ be the set of all sequences $\\{a_{n}\\}_{n\\in\\mathbb{N}}$ of elements of a  $C^*$-algebra $\\mathcal A$ such that the series $\\sum_{n\\in \\mathbb{N}}a_{n}a^*_{n}$ is convergent in $\\mathcal A$. Then $l^{2}(\\mathcal A)$ is a Hilbert $\\mathcal A$-module with respect to the pointwise operations and inner product defined\n\\begin{equation*}\n\\langle \\{a_n\\}_{n\\in \\mathbb N} , \\{b_n\\}_{n\\in \\mathbb N}\\rangle =\\sum_{n\\in  \\mathbb N}a_n b^*_n.\n\\end{equation*}\n\\end{example}\n\\begin{definition}\nLet $E$ be a left $\\mathcal A$-module, an $\\mathcal A$-combination of $x_{1},x_{2},...,x_{n}$ in $E$ is written as follows\n\\begin{equation*}\n \\sum_{i=1}^{n} a_{i}x_{i}=a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}  \\quad {(a_{i} \\in \\mathcal A)}.\n\\end{equation*}\n $x_{1},x_{2},...,x_{n}$ are called $\\mathcal A$-independent if the equation $a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=0$ has exactly one solution, namely $a_{1}=a_{2}=...=a_{n}=0$, otherwise, we say that  $x_{1},x_{2},...,x_{n}$ are $\\mathcal A$-dependent.\\\\The maximum number of elements  in $E$ that are $\\mathcal A$-independent is called the $\\mathcal A$-rank of $E$.\n\\end{definition}\n\\begin{definition}\nLet $\\mathcal A$ be a C*-algebra and $ E$ be a linear space by$\\mathcal A$-rank greater than 1,  which is also a left $\\mathcal A$-module.  We define a function $ \\langle . , .|  .\\rangle  :E\\times E\\times E \\to \\mathcal A$ satisfies  the  following  properties\\\\\n$(T_1)$ $ \\langle x,x|y\\rangle  =0$, if and only if $x=ay$  for $ a\\in\\mathcal A $  \\\\\n$(T_2)$ $ \\langle x , x | y\\rangle  \\geq 0 $ for all $x ,y \\in E$ \\\\\n$(T_3)$ $ \\langle x ,x | y\\rangle  =\\langle y , y | x\\rangle $ for all $x ,y \\in E $ \\\\\n$(T_4)$ $ \\langle x ,y | z\\rangle  =\\langle y , x | z\\rangle ^*$ for all $x ,y ,z \\in E $\\\\\n$(T_5)$ $ \\langle ax ,by | z\\rangle  = a\\langle x , y | z\\rangle  b^*$ for all $x ,y ,z \\in E$  and $ a, b \\in \\mathcal A$ \\\\\n$(T_6)$ $ \\langle x ,\\alpha y | z \\rangle = \\overline{\\alpha}\\langle x , y | z\\rangle $ for all$x ,y \\in E $ and $\\alpha\\in \\mathbb{C} $\\\\\n$(T_7)$ $ \\langle x+y ,z | w\\rangle  =\\langle x , z | w\\rangle  +\\langle y , z | w\\rangle $ for all $x ,y , z ,w \\in E  .$\\\\\nThen the function $ \\langle . , .|  .\\rangle $ is called $ \\mathcal A$-2- inner product  and $ ( E , \\langle . , .|  .\\rangle  ) $ is called  $\\mathcal A$-2-inner product space. \n\\end{definition}\n\\begin{example}\\cite{MS}\nLet $\\mathcal  A $ be a commutative $C^*$-algebra and $E$ be a  pre-Hilbert $\\mathcal A$-module  with inner product $\\langle . , . \\rangle,$  define\n$ \\langle . , .|  .\\rangle $ :$ E\\times E\\times E \\to \\mathcal A$ by \\\\\n\\begin{align*}\n (x ,y ,z)\\longmapsto \\langle x , y | z \\rangle  =\\langle x , y\\rangle \\langle z ,z\\rangle  -\\langle x ,z\\rangle\\langle z ,y\\rangle\n\\end{align*}\nThen  $ ( E , \\langle . , . |  .\\rangle ) $ is a  $\\mathcal A$-2- inner product space.\n\\end{example}\n\\begin{theorem}\nLet $(E,\\langle . , . | .\\rangle )$ be an $ \\mathcal A$-2-inner product space on a  commutative $C^*$-algebra $\\mathcal A$. Then the following inequality holds,\\\\\n\\begin{align*}\n|\\langle x , y | z\\rangle |^{2}=\\langle x , y | z\\rangle \\langle x , y | z\\rangle ^{*}\\leq \\langle x , x | z\\rangle \\langle y , y | z\\rangle \\quad {(x,y,z\\in E).}\n\\end{align*}\n\\begin{proof}\nFor $\\lambda \\in \\mathcal A$ we have \n\\begin{align*}\n0\\leq\\langle \\lambda x-y ,\\lambda x-y|z\\rangle &=\\langle \\lambda x,\\lambda x|z\\rangle -\\langle\\lambda x,y| z\\rangle -\\langle y ,\\lambda x|z\\rangle +\\langle y,y|z\\rangle \\\\&=\\lambda^*\\langle x,x|z\\rangle \\lambda-\\lambda^*\\langle x,y| z\\rangle -\\langle y,x|z\\rangle \\lambda+\\langle y,y|z\\rangle.\n\\end{align*}\nTake $\\lambda=\\langle x,y|z\\rangle (\\langle x,x|z\\rangle+\\varepsilon e)^{-1}$ then\\\\\n\\begin{align*}\n0&\\leq \\langle y,x|z\\rangle (\\langle x,x|z\\rangle +\\varepsilon e)^{-1}\\langle x,x|z\\rangle \\langle x,y|z\\rangle(\\langle x,x|z\\rangle +\\varepsilon e)^{-1}\\\\&-\\langle y,x|z\\rangle (\\langle x,x|z\\rangle +\\varepsilon e)^{-1}\\langle x,y|z\\rangle -\\langle y,x|z\\rangle \\langle x,y|z\\rangle (\\langle x,x|z\\rangle +\\varepsilon e)^{-1}+\\langle y,y|z\\rangle,\n\\end{align*} \nhence,\n$2\\langle y,x|z\\rangle \\langle x,y|z\\rangle \\leq (\\langle x,x|z\\rangle +\\varepsilon e)^{-1}\\langle x,x|z\\rangle \\langle y,x|z\\rangle \\langle x,y|z\\rangle \\\\+\\langle y,y|z\\rangle (\\langle x,x|z\\rangle +\\varepsilon e)\\leq (\\langle x,x|z\\rangle+\\varepsilon e)^{-1}(\\langle x,x|z\\rangle +\\varepsilon e)\\langle y,x|z\\rangle \\langle x,y|z\\rangle +\\langle y,y|z\\rangle (\\langle x,x|z\\rangle +\\varepsilon e)$\nthen by $\\varepsilon\\rightarrow 0$ inequality holds.\n\\end{proof}\n\\end{theorem}\n\\begin{definition}\\cite{TMN}\nLet $E$ be a real vector space that $\\mathcal A$-rank is greater than 1 and\\\\ $p:E\\times E\\to\\mathbb{ R}$ be a function such that\\\\\n$(1)$ $p(x , y) =0$  if and only if $x, y \\in E $ are linearly $\\mathcal A$ - dependent, \\\\\n$(2)$ $p(x , y)=p(y , x)$ for every $x , y \\in E,$\\\\\n$(3)$ $p(\\alpha x, y)=|\\alpha|p(x ,y )$, for every  $x , y\\in E$ and for every $\\alpha\\in\\mathbb{C},$\\\\\n$(4)$ $p(x+y , z )\\leq p(x , z)+p(y , z)$, for every $x , y ,z \\in E.$\\\\\n$(5)$ $P(ax , y )\\leq||a||p(x , y )$ ,  for every $x, y \\in E$ and $a\\in \\mathcal A$,\nThe function $p$ is called an $\\mathcal A$-2-norm.\n\\end{definition}\nIt follows from theorem 1.6 that\n\\begin{corollary}\\cite{TMN}\nLet $E$ be an  $\\mathcal A$-2- inner product space, For $x,z\\in E$ we define $ p(x , z ) = \\sqrt{\\Arrowvert\\Big(\\langle x , x | z\\rangle \\Big)\\Arrowvert}$. Then $ ||\\langle x , y | z\\rangle||\\leq p(x, z )p(y, z )$.\n\\end{corollary}\n\nIn the following theorem, we investigate some properties of an $ \\mathcal A$-2-norm.\n\\begin{theorem}\nLet $(E,\\langle . , . | .\\rangle )$ be an $\\mathcal A$-2-inner product space and $p$ be an  $\\mathcal A$-2-norm,  then \\\\\n$(1)$ $p(x, y) = \\sup\\left\\{\\Arrowvert\\langle x, z|y \\rangle\\Arrowvert;p(z, y)= 1\\right\\}$.\\\\\n$(2)$ $p(x,y+ax)=p(x,y)$ for $a\\in \\mathcal A$.\n\\begin{proof}\n$(1)$ By the Cauchy-schwarz inequality we observe that\n$\\Arrowvert\\langle x, z|y \\rangle\\Arrowvert\\leq p(x, y)p(z, y)\\leq p(x,y),$\nfor every $z\\in E$ such that $p(z, y)\\leq 1$. \nMoreover if $z=\\dfrac{x}{p(x,y)}$ then $p(z,y)=1$ and therefore  $\\Arrowvert\\langle x, z| y\\rangle\\Arrowvert=p(x,y)$.\n\\end{proof}\n\\end{theorem}\nLet $E$ be an $\\mathcal  A$-2- inner product space. A sequence $\\{a_{n}\\}_{n\\in \\mathbb{N}}$ of $E$ is said to be convergent if there exists an element $a\\in E$ such that $\\lim_{n\\rightarrow\\infty}p(a_{n}-a,x)=0$,  for all $x\\in E$. Similarly, we can define a Cauchy sequence in $E$. An $\\mathcal A$-2- inner product space $E$ is called  an  $\\mathcal A$-2- Hilbert  space if it is complete.\\\\\nNow, we give the notion of a frame on a Hilbert $\\mathcal A$-module which is defined in \\cite[definition 3.1]{WJ}.\n\\begin{definition}\\cite{WJ}\nLet $\\mathcal{A}$ be an unital  $C^*$-algebra and $ E$ be a  Hilbert $ \\mathcal{A}$-module.  The sequence $\\{x_{j}\\in~ E | j\\in J\\subseteq\\mathbb{N}\\}$ is called  a frame for $E$ if there exist two  positive elements $A$ and $B$ in real numbers such that\n\\begin{align*}\nA\\langle x,x\\rangle \\leq\\sum_{j\\in J}\\langle x,x_{j}\\rangle \\langle x_{j},x\\rangle \\leq B\\langle x,x\\rangle  \\qquad( x\\in  E).\n\\end{align*}\nThe frame $\\{x_{j}\\}$ is said to be tight frame if $A=B$, and said to be Parseval if $A=B=1$.\\\\\nThe operator $T:E\\rightarrow {l}^2(\\mathcal{A})$ defined by\n\\begin{align*}\nTx=\\{\\langle x,x_{j}\\rangle \\}_{j\\in J}\n\\end{align*}\nis called the analysis operator. The adjoint operator $T^*: {l}^2(\\mathcal{A}) \\rightarrow E$ is given by\n\\begin{align*}\nT^*\\{c_{j}\\}_{j\\in J}=\\sum_{j\\in J}c_{j}x_{j}\n\\end{align*}\n is called the pre-frame operator or the synthesis operator. By composing $T$ and $T^*$, we obtain the frame operator $ S:E\\rightarrow E,$ by\n\\begin{align*}\nS=T^*Tx=\\sum_{j\\in J}\\langle x,x_{j}\\rangle x_{j}.\n\\end{align*}\nAlso from this equation, we have\n\\begin{align*}\nx=\\sum_{j\\in J}\\langle x,S^{-1}x_{j}\\rangle x_{j}  \\qquad( x\\in  E).\n\\end{align*}\n\\end{definition}\nNow  we are ready to define an $\\mathcal A$-2-frame on an  $\\mathcal A$-2- Hilbert space.\n\\section{ $\\mathcal A$-2-frames}\nIn this section we define $\\mathcal A$-2-frames  on $\\mathcal A$-2- Hilbert spaces, and  we give some results about them.\n\\begin{definition}\nLet $(E,\\langle ., . | .\\rangle )$ be an $\\mathcal A$-2- Hilbert space and $\\xi \\in E$. A sequence  $\\{a_{i}\\}_{i\\in \\mathbb{N}}$ of $E$ is called an $\\mathcal A$-2-frame (associated to $\\xi$) if there exist  positive real  numbers $A$ and $B$ such that\n\\begin{align}\\label{2.1}\nA\\langle x,x|\\xi\\rangle \\leq\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi\\rangle \\langle x_{i}, x |\\xi\\rangle \\leq B\\langle x ,x |\\xi \\rangle \\qquad(x\\in  E ).\n\\end{align}\nA sequence satisfying the upper $\\mathcal A$-2-frame condition is called an $\\mathcal A$-2-Bessel sequence, and every $x_{i}$ is $A$- independent to $\\xi$.\n\\end{definition}\n\\begin{proposition}\nLet $\\mathcal A$ be a commutative and  $ (E,\\langle . , .\\rangle )$ be a Hilbert $\\mathcal A$-module and $\\{x_{i}\\}_{i\\in \\mathbb{N}}$ be a frame for $E$. Then  for invertible element $\\langle \\xi, \\xi \\rangle $,  it is  an $\\mathcal A$-2-frame with the standard $\\mathcal A$-2-inner product.\n\\begin{proof}\n\\begin{align*}\n\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi\\rangle \\langle x_{i}, x |\\xi\\rangle &=\\sum_{i\\in \\mathbb{N}}\\langle x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi , x_{i}\\rangle \\langle x_{i} ,x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi\\rangle \\\\&\\leq B\\langle x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi,x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi\\rangle \\\\&\\leq B\\langle \\xi, \\xi \\rangle \\Big(\\langle x , x\\rangle \\langle \\xi , \\xi\\rangle -\\langle x , \\xi\\rangle \\langle \\xi , x\\rangle \\Big)\\\\&=B\\langle \\xi, \\xi \\rangle \\Big(\\langle x , x |\\xi\\rangle \\Big)\\leq ||B\\langle \\xi, \\xi \\rangle ||\\Big(\\langle x , x |\\xi\\rangle \\Big)\n\\end{align*}\nTake $D=||B\\langle \\xi , \\xi\\rangle ||$, the argument for lower bound is similar.\n\\end{proof}\n\\end{proposition}\n\nIn the following proposition, $E$ is a  Hilbert $\\mathcal A$-module in which every closed submodule is orthogonally complemented and $\\langle \\xi, \\xi \\rangle $ is invertible and $L_{\\xi}$ is the subspace generated with $\\xi$.\n\\begin{proposition}\nLet $\\mathcal A$ be a commutative and $ (E,\\langle . , .\\rangle )$ be a Hilbert $\\mathcal A$-module and $\\xi\\in E$. Every $\\mathcal A$-2-frame associated with $\\xi$ is a frame for $L^{\\bot}_{\\xi}$.\n\\begin{proof}\n\\begin{align*}\n\\\\&\\sum_{i\\in \\mathbb{N}}\\langle x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi , x_{i}\\rangle \\langle x_{i} ,x\\langle \\xi \\rangle -\\langle \\xi, x\\rangle \\xi\\rangle \\\\&\\leq B\\langle x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi,x\\langle \\xi, \\xi \\rangle -\\langle \\xi, x\\rangle \\xi\\rangle \\\\&\\leq B\\langle \\xi, \\xi \\rangle \\Big(\\langle x , x\\rangle \\langle \\xi , \\xi\\rangle -\\langle x , \\xi\\rangle \\langle \\xi , x\\rangle \\Big)\n\\end{align*}Then\n\\begin{align*}\nA\\langle x , x \\rangle \\langle \\xi, \\xi\\rangle ^{2}\\leq\\langle \\xi , \\xi\\rangle ^{2}\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i}\\rangle \\langle x_{i} , x\\rangle \\leq B\\langle \\xi , \\xi \\rangle ^{2}\\langle x , x\\rangle   \\qquad(x\\in L^{\\bot}_{\\xi})\n\\end{align*}\nSince $\\langle \\xi , \\xi \\rangle $ is invertible, the  proof is completed.\n\\end{proof}\n\\end{proposition}\nLet $(E,\\langle ., . | .\\rangle )$ be an $\\mathcal A$-2-Hilbert space and $L_{\\xi}$ be the subspace generated with $\\xi$ for a fix element $\\xi$ in $E$. Denote by $\\mathcal M_{\\xi}$ the algebraic complement of $L_{\\xi}$ in $E$. So  $L_{\\xi}\\oplus\\mathcal M_{\\xi}=E$.\nWe define  the semi-inner product $ \\langle . , .\\rangle _{\\xi}$ on $E$ as following\n\\begin{align*}\n \\langle x, z\\rangle _{\\xi}=\\langle x , z |\\xi\\rangle .\n\\end{align*}\nThis semi-inner product induces an inner product on the quotient space $E\/L_{\\xi}$ as\n\\begin{align*}\n\\langle x+L_{\\xi}, z+L_{\\xi}\\rangle _{\\xi}=\\langle x , z \\rangle_{\\xi}.  \\qquad(z,  x\\in  E).\n\\end{align*}\nBy identifying  $E\/L_{\\xi}$ with $\\mathcal M_{\\xi}$ in an obvious way, we obtain an inner product on $\\mathcal M_{\\xi}$.\\\\\nNow if $\\{x_{i}\\}_{i\\in \\mathbb{N}}\\subseteq E$  is an $\\mathcal A$-2- frame associated with $\\xi$ with bounds $A$ and $B$, we can rewrite(2.1) as \n\\begin{align*}\nA\\langle x , x \\rangle _{\\xi}\\leq\\sum_{i\\in \\mathbb{ N}}\\langle x , x_{i}\\rangle \\langle x_{i},x\\rangle \\leq B\\langle x , x\\rangle _{\\xi}     \\qquad(x\\in\\mathcal {M_{\\xi}}).\n\\end{align*}\nThat is, $\\{x_{i}\\}_{i\\in \\mathbb{N}}$ is a frame for  $\\mathcal M_{\\xi}$. Let $ E_{\\xi}$ be the completion of the inner product space  $ E_{\\xi}$, then the sequence  $\\{x_{i}\\}_{i\\in \\mathbb{N}}$ is also a frame for $ E_{\\xi}$. To summarize, we have the following theorem.\n\\begin{theorem}\nLet $(E,\\langle ., . | .\\rangle )$ be an $\\mathcal A$-2-Hilbert space. Then  $\\{x_{i}\\}_{i\\in \\mathbb{N}}\\subseteq E$ is  an $\\mathcal A$-2-frame associated with $\\xi$ if and only if it is a frame for the Hilbert space  $ E_{\\xi}$.\n\\end{theorem}\n\\begin{lemma}\nLet  $\\{x_{i}\\}_{i\\in \\mathbb{N}}$ be an $\\mathcal A$-2-Bessel sequence in $E$. Then the $\\mathcal A$-2-pre frame operator $T:l^2(\\mathcal A)\\rightarrow E_{\\xi}$\ndefined by\n\\begin{align*}\nT_{\\xi}\\{c_{i}\\}=\\sum_{i\\in \\mathbb{N}}c_{i}x_{i}\n\\end{align*}\nis well-defined and bounded.\n\\begin{proof}\n\\begin{align*}\n\\| \\sum_{i=1}^{n} c_{i}x_{i}- \\sum_{i=1}^{m} c_{i}x_{i} , \\xi \\| ^{2}&=sup\\{\\| \\langle \\sum_{i=m+1}^{n} c_{i}x_{i} , y | \\xi \\rangle  \\|^{2}  , y\\in E_{\\xi} ,\\|  y , \\xi  \\|=1\\}\\\\&\\leq sup\\{\\| \\langle  \\sum_{i=m+1}^{n} \\langle x_{i} , y | \\xi \\rangle \n\\langle y , x_{i} | \\xi \\rangle  \\|  , y\\in E_{\\xi} ,\\|  y , \\xi  \\|=1\\} \\| \\sum_{i=m+1}^{n} c_{i}c^*_{i} \\|\\\\& \\leq \\| \\sum_{i=m+1}^{n} c_{i}c^*_{i} \\| \\| B \\| .\n\\end{align*}\nSince $B$ is the upper bound of $\\{x_{i}\\}$ this implies that $\\sum_{i=1}^{\\infty}c_{i}x_{i}$ is well defined as an element of $E_{\\xi}$. Moreover if $\\{c_{i}\\}$ is a sequence in $ l^2(\\mathcal A)$, then an argument as above shows that\n\\begin{align*}\n\\| T_{\\xi}(\\{c_{i}\\}) \\| \\leq \\sqrt{\\| B \\| } \\| \\sum_{i \\in \\mathbb{N}}c_{i}c^*_{i} \\|\n\\end{align*}\nIn particular, $\\| T_{\\xi}\\| \\leq  \\sqrt{\\| B \\| }$.\n\\end{proof}\n\\end{lemma}\nWe have $\\langle x , T(c_{j}) | \\xi \\rangle=\\langle x , T(c_{j}) \\rangle_{\\xi}=\\langle x ,\\sum c_{j}x_{j} \\rangle_{\\xi}=\\sum \\langle x ,x_{j} \\rangle_{\\xi}c_{j}^*=\\langle {\\langle x, x_{j} \\rangle_{\\xi}} , {c_{j}} \\rangle$\nNext, we can compute $T^*_{\\xi}$, the adjoint of  $T_{\\xi}$ as\n\\begin{align*}\nT^*_{\\xi}: E_{\\xi}\\rightarrow l^2(\\mathcal A);    \\qquad T^*_{\\xi}x=\\{\\langle x , x_{i}|\\xi\\rangle \\}_{i\\in \\mathbb{N}}.\n\\end{align*}\n $T^*_{\\xi}$ is well-defined and bounded, because\n\\begin{align*}\n\\| T^*_{\\xi}(x) \\|^{2}=\\| \\{\\langle x , x_{i} ,|\\xi \\rangle \\}_{i \\in \\mathbb{N}} \\|^{2}=\\| \\sum_{i \\in \\mathbb{N}}\\langle x , x_{i} ,|\\xi \\rangle \\langle x_{i} , x | \\xi \\rangle  \\| \\leq \\|B\\| \\|x , \\xi \\|\n\\end{align*}\nThat implies $\\| T^*_{\\xi} \\| \\leq \\sqrt {\\| B \\|}$.\n\\begin{definition}\\label{2.2}\nLet $ \\{x_{i}\\}_{i\\in \\mathbb{N}}$ be an $\\mathcal A$-2-frame  associated to $\\xi$   with bounds $A$ and $B$ in an $\\mathcal A$-2- Hilbert space $E$. The operator $S_{\\xi}:E_{\\xi}\\rightarrow E_{\\xi}$ defined by\n\\begin{align}\nS_{\\xi}x=\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi\\rangle x_{i}.\n\\end{align}\n is called the $\\mathcal A$-2-frame operator for $ \\{x_{i}\\}_{i\\in \\mathbb{N}}$.\n\\end{definition}\nIn the next theorem, we investigate some properties of $S_{\\xi}.$\n\\begin{theorem}\nLet $ \\{x_{i}\\}_{i\\in \\mathbb{N}}$ be an $\\mathcal A$-2-frame  associated to $\\xi$ for an $\\mathcal A$-2-Hilbert space $(E ,\\langle . ,. | \\rangle )$ with $\\mathcal A$-2-frame operator $ S_{\\xi}$. Then $ S_{\\xi}$ is bounded,  invertible, self-adjoint, and positive.\n\\begin{proof}\nIt is clear that $ S_{\\xi}=T_{\\xi}T^*_{\\xi}$ is self adjoint and\n\\begin{align*}\n\\| S_{\\xi} \\|=\\| T_{\\xi}T^*_{\\xi} \\|=\\| T_{\\xi} \\|^{2} \\leq \\| B \\|.\n\\end{align*}\nWe can conclude the boundedness of $S_{\\xi}$ directly\n\\begin{align*}\n\\| S_{\\xi}(x) , \\xi \\|^{2}&=sup\\{\\| \\langle  S_{\\xi}(x) , y | \\xi \\rangle  \\|^{2}  , y\\in E_{\\xi} ,\\|  y , \\xi  \\|=1\\}\\\\&= sup\\{\\| \\langle \\sum_{i=1}^{\\infty}\\langle x , x_{i} | \\xi \\rangle x_{i} , y | \\xi \\rangle  \\|^{2} , y\\in E_{\\xi} ,\\|  y , \\xi  \\|=1\\}\\\\& \\leq sup\\{\\| \\sum_{i=1}^{\\infty}\\langle x , x_{i} | \\xi \\rangle \\langle x_{i} , x | \\xi \\rangle \\| \\| \\sum_{i=1}^{\\infty}\\langle y , x_{i} | \\xi \\rangle \\langle x_{i} , y | \\xi \\rangle \\|  ,\\|  y , \\xi  \\|=1 \\}\\\\& \\leq \\| B \\|^{2} \\| \\langle  x , x | \\xi \\rangle \\|\n\\end{align*}\nThe inequality (\\ref{2.1}) means that \n\\begin{align*}\nA\\langle x,x \\rangle _{\\xi}\\leq \\langle S_{\\xi}(x), x \\rangle _{\\xi}\\leq B\\langle x ,x \\rangle _{\\xi} \n\\end{align*}\n$ S_{\\xi} $ is a positive element in the set of all bounded operators on the Hilbert space $E_{\\xi}$.\n\\end{proof}\n\\end{theorem}\nBy the definition of $S_{\\xi}$ we get the following results.\n\\begin{corollary}\nLet $ \\{x_{i}\\}_{i\\in \\mathbb{N}}$ be an $\\mathcal A$-2-frame in an $\\mathcal A$-2-Hilbert space $(E ,\\langle . ,. | \\rangle )$ with frame operator  $ S_{\\xi}$. Then each $x\\in E_{\\xi}$ has an expansion of  the following\n\\begin{align*}\nx= S S_{\\xi}^{-1}x=\\sum_{i\\in \\mathbb{N}}\\langle  S_{\\xi}^{-1}x , x_{i} |\\xi\\rangle x_{i}.\n\\end{align*}\n\\end{corollary}\n\\begin{corollary}\nLet $\\xi$ and $\\eta$ be $\\mathcal A$-independent and $ \\{x_{i}\\}_{i\\in \\mathbb{N}}$ be an $\\mathcal A$-2-frame associated with $\\xi$ and $\\eta$,  and for $x\\in E_{\\xi}\\cap E_{\\eta}$, the operators $ S_{\\xi}, E_{\\eta}:E_{\\xi}\\cap E_{\\eta}\\rightarrow E_{\\xi}\\cap E_{\\eta}$ defined by,\n\\begin{align*}\nS_{\\xi}x=\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi\\rangle x_{i},\\\\\nS_{\\eta}x=\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\eta\\rangle x_{i}.\n\\end{align*}\nThen we have $\\langle S_{\\eta}x, x | \\xi \\rangle =\\langle S_{\\xi} x, x | \\eta \\rangle^*$.\n\\begin{proof}\n\\begin{align*}\n\\langle S_{\\eta}x, x | \\xi \\rangle &=\\langle \\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\eta\\rangle x_{i}, x | \\xi \\rangle\\\\& =\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\eta\\rangle \\langle x_{i}, x | \\xi \\rangle\\\\&=(\\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi \\rangle \\langle x_{i}, x | \\eta \\rangle )^*\\\\&=\\langle \\sum_{i\\in \\mathbb{N}}\\langle x , x_{i} |\\xi\\rangle x_{i}, x | \\eta \\rangle^*\\\\&=\\langle S_{\\xi} x, x | \\eta \\rangle^*.\n\\end{align*}\n\\end{proof}\n\\end{corollary}\n\\section{Tensor product of $\\mathcal A$-2-Frames }\nLet $\\mathcal A$ and $\\mathcal B$ be $C^*$-algebras, $E$ an $\\mathcal A$-2-Hilbert space and $F$ be a $\\mathcal B$-2-Hilbert space. We take $\\mathcal A\\otimes \\mathcal B$ as the completion of $\\mathcal A\\otimes_{alg} \\mathcal B$ with the spatial norm. Hence  $\\mathcal A\\otimes \\mathcal B$ is a $C^*$-algebra and for every $a\\in \\mathcal A $ and $b\\in \\mathcal B$ we have $\\|a\\otimes b\\|=\\|a\\|\\|b\\|$. The algebraic tensor product $ E\\otimes_{alg} F$ is a pre-Hilbert $\\mathcal A\\otimes \\mathcal B$-module with module action \n\\begin{align*}\n(a\\otimes b)(x\\otimes y)=ax\\otimes by     \\qquad (a\\in A, b\\in B, x\\in E, y\\in F )\n\\end{align*}\nand  $\\mathcal A\\otimes \\mathcal B$-valued 2-inner product\n\\begin{align*}\n\\langle x_{1}\\otimes y_{1}, x_{2}\\otimes y_{2} | \\xi \\otimes \\eta  \\rangle=\\langle x_{1},  x_{2} | \\xi  \\rangle \\otimes\\langle y_{1},  y_{2} | \\eta \\rangle  \\qquad (x_{1},  x_{2}, \\xi \\in E, y_{1},  y_{2}, \\eta \\in F).\n\\end{align*}\nThe $\\mathcal A\\otimes \\mathcal B$-2-norm on $ E\\otimes F$ is defined by\n\\begin{align*}\n\\|x_{1}\\otimes x_{2}, y_{1}\\otimes y_{2}\\|=\\|x_{1}, y_{1}\\|_{\\mathcal A}\\|x_{2}, y_{2}\\|_{\\mathcal B}  \\qquad (x_{1}, y_{1}\\in E, x_{2}, y_{2}\\in F ).\n\\end{align*}\nwhere $\\|. , . \\|_{\\mathcal A}$ and $\\|. , . \\|_{\\mathcal B}$ are norms generated by $\\langle . , . | . \\rangle_{\\mathcal A}$ and $\\langle . , . | . \\rangle_{\\mathcal B}$ respectively. The space $ E\\otimes F$ is complete with the above 2-inner product. Therefore, the space $ E\\otimes F$ is an  $\\mathcal A\\otimes \\mathcal B$-2-Hilbert space.\\\\\nThe following definition is the extension of (2.1) to the sequence $\\{x _{i}\\otimes y_{i}\\}_{i\\in \\mathbb{N}}$.\n\\begin{definition}\nLet $\\{x _{i}\\}$ and $\\{y _{i}\\}$ be two sequences in $\\mathcal A$-2-Hilbert space $E$ and $\\mathcal B$-2-Hilbert space $F$,  respectively. Then, the sequence $\\{x _{i}\\otimes y_{i}\\}_{i\\in \\mathbb{N}}$ is said to be a tensor product of $\\mathcal A\\otimes \\mathcal B$-2-frame for the tensor product of $\\mathcal A\\otimes \\mathcal B$-2-Hilbert space $ E\\otimes F$ associated to $\\xi \\otimes \\eta$ if there exist two constants $0<A\\le B<\\infty$ such that\n\\begin{align*}\nA\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle\\leq \\sum_{i,j}\\langle x\\otimes y, x_{i}\\otimes y_{j} | \\xi \\otimes \\eta  \\rangle\\langle x_{i}\\otimes y_{j}, x\\otimes y | \\xi \\otimes \\eta  \\rangle\\leq B\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle\n\\end{align*} \nfor all $x\\in  E, y\\in F$.\n\nThe numbers $A$ and $B$ are called lower and upper frame bounds of the tensor product of  $\\mathcal A\\otimes \\mathcal B$-2-frame respectively.\n\\end{definition}\n\\begin{lemma}\nLet  $\\{x _{i}\\}_{i\\in I}$ be an $\\mathcal A$-2-frame for $E$ with frame bounds $A$ and $B$ associated to $\\xi$ and let $\\{y _{i}\\}_{j\\in J}$  be a  $\\mathcal B$-2-frame for $F$ with frame bounds $C$ and $D$ associated with $\\eta$. Then  $\\{x _{i}\\otimes y_{i}\\}_{i\\in I, j\\in J}$ is an $\\mathcal A\\otimes \\mathcal B$-2-frame for $ E\\otimes F$ with frame bounds $AC$ and $BD$ associated with $\\xi\\otimes \\eta $,  if  $\\{x _{i}\\}_{i\\in I}$ and $\\{y _{i}\\}_{j\\in J}$ are tight or Parseval frames, then so is  $\\{x _{i}\\otimes y_{i}\\}_{i\\in I, j\\in J}$ .\n\\end{lemma}\n\\begin{proof}\nLet $x\\in E$ and $y\\in F$. Then we have\n\\begin{align} \nA\\langle x, x |\\xi \\rangle\\leq \\sum_{i\\in I}\\langle x, x_{i} |\\xi \\rangle\\langle x_{i}, x |\\xi \\rangle\\leq B\\langle x, x |\\xi \\rangle\n\\end{align}\n\\begin{align} \nC\\langle y, y |\\eta \\rangle\\leq \\sum_{j\\in J}\\langle y, y_{i} |\\eta \\rangle\\langle y_{i}, y |\\eta \\rangle\\leq B\\langle y, y |\\eta \\rangle\n.\\end{align} \nWe know if $a$, $b$ are hermitian elements of $\\mathcal A$ and $a\\leq b$, then for every positive element $x$ of $\\mathcal B$, we have $a\\otimes x \\leq b\\otimes x$. Therefore\n\\begin{align*}\nA\\langle x, x |\\xi \\rangle\\otimes \\langle y, y |\\eta \\rangle\\leq \\sum_{i\\in I}\\langle x, x_{i} |\\xi \\rangle\\langle x_{i}, x |\\xi \\rangle \\otimes \\langle y, y |\\eta \\rangle\\leq B\\langle x, x |\\xi \\rangle\\otimes \\langle y, y |\\eta \\rangle.\n\\end{align*}\nNow by (3.2), we have\n\\begin{align*} \nAC\\langle x, x |\\xi \\rangle\\otimes \\langle y, y |\\eta \\rangle&\\leq \\sum_{i\\in I} \\sum_{j\\in J}\\langle x, x_{i} |\\xi \\rangle\\langle x_{i}, x |\\xi \\rangle \\otimes \\langle y, y_{i} |\\eta \\rangle\\langle y_{i}, y |\\eta \\rangle \\\\&\\leq  B\\langle x, x |\\xi \\rangle\\otimes \\sum_{j\\in J}\\langle y, y_{i} |\\eta \\rangle\\langle y_{i}, y |\\eta \\rangle\\\\& \\leq  BD\\langle x, x |\\xi \\rangle\\otimes \\langle y, y |\\eta \\rangle.\n\\end{align*}\nConsequently, we have\n\\begin{align*} \nAC\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle&\\leq \\sum_{i,j}\\langle x\\otimes y, x_{i}\\otimes y_{j} | \\xi \\otimes \\eta  \\rangle\\langle x_{i}\\otimes y_{j}, x\\otimes y | \\xi \\otimes \\eta  \\rangle\\\\&\\leq BD\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle.\n\\end{align*}\nFrom these inequalities, it follows that for all $z=\\sum_{k=1}^{n}u_{k}\\otimes v_{k} \\in E\\otimes _{alg} F$ \n\\begin{align*} \nAC\\langle z, z | \\xi \\otimes \\eta  \\rangle&\\leq \\sum_{i,j}\\langle z, z | \\xi \\otimes \\eta  \\rangle\\langle x_{i}\\otimes y_{j}, z | \\xi \\otimes \\eta  \\rangle\\\\&\\leq BD\\langle z, z | \\xi \\otimes \\eta  \\rangle.\n\\end{align*} \nHence it  holds for all $z\\in E\\otimes  F$.\n\\end{proof}\nIf take $\\mathcal A=\\mathbb{C}$ and define $\\mathbb{C}\\otimes \\mathcal B$-2-product by \n\\begin{align}\\label{3.3}\n\\langle x_{1}\\otimes y_{1}, x_{2}\\otimes y_{2} | \\xi \\otimes \\eta  \\rangle=\\langle x_{1},  x_{2} | \\xi  \\rangle \\langle y_{1},  y_{2} | \\eta \\rangle  \\qquad (x_{1},  x_{2}, \\xi \\in E, y_{1},  y_{2}, \\eta \\in F).\n\\end{align}\nWe have the following proposition.\n\\begin{proposition}\nLet $\\{x _{i}\\otimes y_{i}\\}_{i\\in I, j\\in J}$ be a $\\mathbb{C}\\otimes \\mathcal B$-2-frame for $ E\\otimes F$ associated with $\\xi\\otimes \\eta $, then $\\{y _{i}\\}_{j\\in J}$  is a  $\\mathcal B$-2-frame for $F$ associated with $\\eta$. \n\\begin{proof}\nSuppose that $\\{x _{i}\\otimes y_{i}\\}_{i\\in I, j\\in J}$ is a $\\mathbb{C}\\otimes \\mathcal B$-2-frame for $ E\\otimes F$ associated to $\\xi\\otimes \\eta $. Then for $x\\otimes y\\in  E\\otimes F-\\{0\\otimes0\\}$, we have\n\\begin{align*}\nA\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle\\leq \\sum_{i,j}\\langle x\\otimes y, x_{i}\\otimes y_{j} | \\xi \\otimes \\eta  \\rangle\\langle x_{i}\\otimes y_{j}, x\\otimes y | \\xi \\otimes \\eta  \\rangle\\leq B\\langle x\\otimes y, x\\otimes y | \\xi \\otimes \\eta  \\rangle.\n\\end{align*} \nBy using (\\ref{3.3}) the above equation becomes\n\\begin{align*} \nA\\langle x, x |\\xi \\rangle \\langle y, y |\\eta \\rangle&\\leq \\sum_{i\\in I} |\\langle x, x_{i} |\\xi \\rangle|^2  \\sum_{j\\in J}\\langle y, y_{j} |\\eta \\rangle\\langle y_{j}, y |\\eta \\rangle \\\\& \\leq  B\\langle x, x |\\xi \\rangle \\langle y, y |\\eta \\rangle.\n\\end{align*}\nThis gives\n\\begin{align*} \n\\frac {A\\langle x, x |\\xi \\rangle}{ \\sum_{i\\in I} |\\langle x, x_{i} |\\xi \\rangle|^2} \\langle y, y |\\eta \\rangle&\\leq  \\sum_{j\\in J}\\langle y, y_{j} |\\eta \\rangle\\langle y_{j}, y |\\eta \\rangle \\\\& \\leq \\frac{ B\\langle x, x |\\xi \\rangle}{ \\sum_{i\\in I} |\\langle x, x_{i} |\\xi \\rangle|^2} \\langle y, y |\\eta \\rangle.\n\\end{align*}\nTherefore \n\\begin{align*} \nA_{1} \\langle y, y |\\eta \\rangle \\leq  \\sum_{j\\in J}\\langle y, y_{j} |\\eta \\rangle\\langle y_{j}, y |\\eta \\rangle \\leq B_{1} \\langle y, y |\\eta \\rangle. \\qquad (\\forall y\\in F)\n\\end{align*}\nWhere $A_{1}=\\frac {A\\langle x, x |\\xi \\rangle}{ \\sum_{i\\in I} |\\langle x, x_{i} |\\xi \\rangle|^2}$ and $B_{1}= \\frac{ B\\langle x, x |\\xi \\rangle}{ \\sum_{i\\in I} |\\langle x, x_{i} |\\xi \\rangle|^2}.$\n\\end{proof}\n\\end{proposition}\n\\begin{remark} \nIf the sequences $\\{x_{i}\\}_{i\\in I}$,  $\\{y_{j}\\}_{j\\in J} $ and$ \\{x_{i}\\otimes y_{j}\\}_{i\\in I,j\\in J}$ are 2-frames for Hilbert spaces $E_{\\xi}$, $F_{\\eta}$ and $(E\\otimes F)_{\\xi \\otimes \\eta}$ respectively and $S_{\\xi}$, $S_{\\eta}$ and $S_{xi \\otimes \\eta}$ are frame operators of above frames respectively, then from (\\ref{2.2}), we have the following\n\\begin{align*} \nS_{\\xi}x=\\sum_{i\\in I}\\langle x , x_{i} |\\xi\\rangle x_{i},\nS_{\\eta}y=\\sum_{j\\in J}\\langle y , y_{j} |\\eta\\rangle y_{j}.\n\\end{align*}\n\\begin{align*}\nS_{\\xi\\otimes \\eta}(x\\otimes y)=\\sum_{i\\in I ,j\\in J}\\langle x\\otimes y , x_{i}\\otimes y_{j} |\\xi \\otimes \\eta\\rangle( x_{i}\\otimes y_{j}),( x\\in E, y\\in F, x\\otimes y\\in E\\otimes F )\n\\end{align*}\n\\end{remark}\n\\begin{theorem}\nIf the sequences $\\{x_{i}\\}_{i\\in I}$,  $\\{y_{j}\\}_{j\\in J} $ and$ \\{x_{i}\\otimes y_{j}\\}_{i\\in I,j\\in J}$ are 2-frames for Hilbert spaces $E_{\\xi}$, $F_{\\eta}$ and $(E\\otimes F)_{\\xi \\otimes \\eta}$, respectively and $S_{\\xi}$, $S_{\\eta}$ and $S_{\\xi \\otimes \\eta}$ are frame operators respectively, then $S_{\\xi \\otimes \\eta}=S_{\\xi}\\otimes S_{\\eta}$.\n\\begin{proof}\nFor $x\\otimes y\\in E\\otimes F$, we have\n\\begin{align*}\nS_{\\xi\\otimes \\eta}(x\\otimes y)&=\\sum_{i\\in I ,j\\in J}\\langle x\\otimes y , x_{i}\\otimes y_{j} |\\xi \\otimes \\eta\\rangle( x_{i}\\otimes y_{j})\\\\&=\\sum_{i\\in I ,j\\in J}\\langle x, x_{i} | \\xi \\rangle_{\\mathcal A} \\langle y , y_{j} |\\eta \\rangle_{\\mathcal B}(x_{i}\\otimes y_{j})\\\\&=\\sum_{i\\in I}\\langle x, x_{i} | \\xi \\rangle_{\\mathcal A}x_{i}\\otimes \\sum_{j\\in J}\\langle y, y_{j} | \\eta \\rangle_{\\mathcal B}y_{i}\\\\&=S_{\\xi}x\\otimes S_{\\eta}y=(S_{\\xi}\\otimes S_{\\eta})(x\\otimes y).\n\\end{align*}\nHence $S_{\\xi \\otimes \\eta}=S_{\\xi}\\otimes S_{\\eta}$.\n\\end{proof}\n\\end{theorem}\n\\begin{lemma}\nSuppose that $\\{x_{i}\\}_{i\\in I}$ is a sequence in $\\mathcal A$-2-Hilbert space $E$, with $x=\\sum_{i\\in I}\\langle x, x_{i} | \\xi \\rangle x_{i}$ holds for all $x\\in E$, then  $\\{x_{i}\\}_{i\\in I}$ is an $\\mathcal A$-2-normalized tight frame for $E$.\n\\begin{proof}\n$x=\\sum_{i\\in I}\\langle x, x_{i} | \\xi \\rangle x_{i}$, so $\\langle x , x | \\xi \\rangle=\\langle \\sum_{i\\in I}\\langle x, x_{i} | \\xi \\rangle x_{i} , x | \\xi \\rangle$, hence $\\Longrightarrow \\langle x , x | \\xi \\rangle=\\sum_{i\\in I}\\langle x, x_{i} | \\xi \\rangle\\langle x_{i} , x | \\xi \\rangle.$\n\\end{proof}\n\\end{lemma}\n\\begin{corollary}\nAssume that  $\\{x_{i}\\otimes y_{j}\\}_{i\\in I, j\\in J}$ is a sequence in $\\mathcal A\\otimes \\mathcal B$-2-Hilbert space $E\\otimes F$ and $x\\otimes y=\\sum_{i\\in I ,j\\in J}\\langle x\\otimes y , x_{i}\\otimes y_{j} |\\xi \\otimes \\eta\\rangle( x_{i}\\otimes y_{j})$, for all $x\\in E, y\\in F$. Then $\\{x_{i}\\otimes y_{j}\\}_{i\\in I, j\\in J}$ is an $\\mathcal A\\otimes \\mathcal B$-2-normalized tight frame for $E\\otimes F$.\n\\begin{proof}\nBy lemma (3.6) is clear.\n\\end{proof}\n\\end{corollary}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{conceptual_illustration8.pdf}\n\\caption{A high-level overview of our proposed method: the CA-UReID method disentangles camera-agnostic feature from the original feature in a explicit way. By jointly optimized with the camera-aware contrastive learning, our method can effectively reduce the intra-class distance and increase the inter-class distance with high matching accuracy.}\n\\label{fig_conception}\n\\vspace{-10pt}\n\\end{figure}\n\nPerson re-identification (ReID) aims at retrieving images of the same person across cameras with non-overlapping views, which has important applications in the field of intelligent surveillance and public security. \nIn the past few years, researchers pay more and more attention to this task and make great research progress.\nHowever, existing high-performance methods mainly use the supervised learning framework with identities' annotation and the process of building a dataset is time-consuming and laborious.\nBesides, the supervised models need to be retrained in a new scenario, which have poor scalability and are not conducive to the practical deployment.\nTherefore, some researchers recently turn their attention to unsupervised person ReID (UReID)~\\cite{DBLP:conf\/aaai\/HuangPJLX20, DBLP:conf\/nips\/Ge0C0L20,DBLP:journals\/corr\/abs-2103-16364, DBLP:conf\/aaai\/0001LHG021}, which has great potential value to get rid of dependence on manual annotation and become an emerging research hotspot.\n\nExisting unsupervised person ReID methods are mainly divided into two categories: unsupervised domain adaptive (UDA) methods and purely unsupervised learning (USL) methods.\nUDA methods~\\cite{DBLP:conf\/aaai\/HuangPJLX20,DBLP:conf\/iclr\/GeCL20,DBLP:conf\/nips\/Ge0C0L20,DBLP:conf\/aaai\/ZhengLZZZ21} first train the model on a supervised source-domain dataset to make the model have a certain discriminant ability for identities.\nThen the training is carried out in the target domain without annotation to transfer the ability.\nHowever, the performance of UDA methods is greatly affected by the quality and size of the annotated source domain datasets, which constrains their scalability.\nIn contrast, the purely unsupervised methods~\\cite{DBLP:conf\/aaai\/0001LHG021,DBLP:journals\/corr\/abs-2103-16364} that requires only the unlabeled target dataset is more flexible. \nUSL methods mainly rely on clustering to generate pseudo labels for supervised learning in the target dataset with the strategy of contrastive learning.\nHowever, most USL methods~\\cite{DBLP:conf\/eccv\/LiZG18, DBLP:conf\/bmvc\/ChenZG18, DBLP:conf\/iccv\/WuLYLLL19} only focus on improving the differences between classes and ignore the intra-class differences caused by camera style variance, including camera property and environment information of cameras, etc.\nSeveral methods~\\cite{DBLP:journals\/tist\/TianTLTZF21,DBLP:conf\/cvpr\/XuanZ21, DBLP:conf\/aaai\/0001LHG021} that consider the influence of cameras are complicated and indirect.\n\nBased on the above observations, we propose a novel purely unsupervised learning method, the camera-aware style separation and contrast learning network (CA-UReID), which directly disentangles camera style at the level of feature map and utilize camera-aware contrastive learning to learn more discriminative feature for each identity (Fig.\\ref{fig_conception}).\nSpecifically, we first design a camera style separation (CSS) module that divides the extracted features into camera-specific and camera-agnostic parts with attention mechanism, so as to alleviate the impact of camera style variance on performance.\nIn the camera-specific branch, the disentangled attention module captures the camera style information under the guidance of the proposed camera separation loss.\nThen, by using complementary attention maps, the camera-agnostic branch can enforce the network to focus on the persons, which extracts more camera-invariant features with the compact intra-class distribution.  \nAfter the CSS module, we propose a camera-aware contrastive center (CACC) loss into the optimization constraints to further narrow the gap between cameras within the class and increase the feature distance of different classes.\nMoreover, our CACC loss considers the constraints on the sample set under the same camera rather than individual samples, which is verified by us as a more stable and efficient way and is not easy to be affected by noise samples.\nThrough joint optimization with the designed camera style separation module and camera-aware contrastive center loss, our method can learn more robust features of identities for the final retrieval.\n\n\\vspace{2pt}\nOur main contributions are summarized as follows:\n\\vspace{-5pt}\n\\begin{itemize}\n  \\item We propose a novel camera style separation module to explicitly disentangle camera-specific information from feature maps and reduce intra-class variance.\n  \n  \\vspace{-5pt}\n  \\item We design a more robust camera-aware contrastive center loss, which further enhances camera-agnostic features at center level to get discriminative embeddings.\n  \n  \\vspace{-5pt}\n  \\item Extensive experiments on mainstream datasets (Market1501~\\cite{DBLP:conf\/iccv\/ZhengSTWWT15} and DukeMTMC-reID~\\cite{DBLP:conf\/eccv\/RistaniSZCT16}) show superior performance of our proposed CA-UReID method.\n\\end{itemize}\n\n\n\\section{Related Work}\n\n\\textbf{Unsupervised Person Re-identification.}\nUnsupervised person ReID methods can be categorized into the unsupervised domain adaptive (UDA) methods and the purely unsupervised learning (USL) methods.\nThe main difference between them is whether to use labeled source domain datasets.\nAs for the UDA methods, some of them use GAN~\\cite{DBLP:conf\/cvpr\/Deng0YK0J18,DBLP:conf\/cvpr\/WeiZ0018} for style transfer to narrow the gap between the source domain and target domain.\nSome other methods are designed to generate more reliable pseudo labels: \nSpCL~\\cite{DBLP:conf\/nips\/Ge0C0L20} introduces a cluster reliability criterion to retain reliable clusters and disassemble unreliable clusters to outlier instances.\nDAAM~\\cite{DBLP:conf\/aaai\/HuangPJLX20} assigns different weights to samples based on their distance from the class centers. \nUNRN~\\cite{DBLP:conf\/aaai\/ZhengLZZZ21} measured the reliability of pseudo labels by inconsistencies between the teacher network and student network.\nCompared with the UDA method, the USL method is more flexible and promising without manual annotations.\nSome approaches explore special clustering methods, such as k-NN~\\cite{DBLP:conf\/eccv\/LiZG18,DBLP:conf\/bmvc\/ChenZG18} and graphs~\\cite{DBLP:conf\/iccv\/WuLYLLL19}, to generate pseudo-labels. \nMMCL~\\cite{DBLP:conf\/cvpr\/WangZ20a} uses a memory-based multi-label classification loss to update the model. \nCycAs~\\cite{DBLP:conf\/eccv\/WangZZLSLW20} learns pedestrian embedding from original videos to mine temporal information.\nMost USL methods focus on the distance between classes but often ignore the intra-class variance caused by inconsistent camera styles. \nIn contrast, our USL-based CA-UReID method considers the effects of camera style, which effectively improves the consistency within the class.\n\n\\vspace{4pt}\n\\noindent\\textbf{Research of Camera Style in UReID.}\nEliminating the effects of camera style has recently been highlighted by a few unsupervised ReID methods.\nCIDC~\\cite{DBLP:journals\/tist\/TianTLTZF21} method transfers samples to each camera via StarGAN~\\cite{DBLP:conf\/cvpr\/ChoiCKH0C18}.\nSome other methods use intra-camera and inter-camera combined training. \nIn the intra-camera training stage, a classifier is set under each camera, and all the classifiers share a backbone network. \nIn the inter-camera training stage, existing methods design different strategies to unify the knowledge learned from different cameras:\nIICS~\\cite{DBLP:conf\/cvpr\/XuanZ21} uses a combination of Instance Normalization (IN) and Batch Normalization (BN) to improve the generalization ability of classifiers. \nCAP~\\cite{DBLP:conf\/aaai\/0001LHG021} designs camera-based clusters to pull positive clusters and push nearest negative proxies. \nMetaCam~\\cite{DBLP:conf\/cvpr\/YangZLCLLS21} uses meta-learning to learn robust models for cameras. \nMost of these methods have complex training processes, and their way of eliminating the influence of camera style is indirect.\nDifferent from them, our proposed CA-UReID method separates camera style in an explicit way by disentangling camera-specific information in the feature space. \nMoreover, compared with the existing CAP~\\cite{DBLP:conf\/aaai\/0001LHG021} and ICE~\\cite{DBLP:journals\/corr\/abs-2103-16364} methods, \nour designed camera-aware contrastive loss considers the overall distribution in each camera-aware cluster, which is more robust with better performance. \n\n\n\\section{Method}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{overall_structure9.pdf}\n\\vspace{-18pt}\n\\caption{\nThe framework of our proposed method.\n(a) The two-branch camera style separation module, including the camera-specific branch and the camera-agnostic branch.\n(b) The pipeline of calculating the camera-aware contrastive center loss.\n}\n\\vspace{-10pt}\n\\label{fig_overall_structure}\n\\end{figure*}\n\nIn this section, we introduce our camera-aware style separation and contrastive learning (CA-UReID) method in detail.\nIn Sec.~\\ref{subsec_overview}, we first describe the overall architecture of CA-UReID, including the structure of network and the overall objective function during the training.\nThen we introduce our proposed camera style separation module and camera-aware contrastive center loss in the next two subsections.\n\n\\subsection{Overall Structure} \\label{subsec_overview}\nAs shown in Fig.~\\ref{fig_overall_structure}, our method follows the memory bank-based framework, constructing an unsupervised learning network.\nThe overall process of CA-UReID mainly includes three stages: identities' feature extraction, two-branch camera style separation, and self-supervised contrastive learning.\n\nIn the feature extraction stage, let $\\mathcal{X}=\\{x_1,x_2,\u2026,x_N\\}$ denote the person ReID dataset, and $N$ is the number of samples.\nSpecifically, given an input sample $x_i$, we use ResNet-50 as the backbone to extract the corresponding feature map $F_{x_i}\\in \\mathbb{R}^{H \\times W \\times C}$, \nwhere $H$, $W$, $C$ denote height, width and channel number respectively. \nIn the stage of two-branch camera style separation, the designed camera separation (Cam-Sep) attention module is used to disentangle the camera-specific feature map $F_{x_i}^{sp}$ and camera-agnostic feature map $F_{x_i}^{ag}$ from $F_{x_i}$.\nThe $F_{x_i}^{sp}$ is fed into the camera-specific branch under the constraints of the proposed camera separation loss $\\mathcal{L}_{sep}$, and we use the $F_{x_i}^{ag}$ to generate the camera-agnostic 2048-dim feature embedding $f_i$ with the global average pooling (GAP) and batch normalization (BN) operations.\n\nDuring the last stage of self-supervised contrastive learning, as shown in the upper part of Fig.~\\ref{fig_overall_structure}, our method follows the framework based on memory bank like some previous methods~\\cite{DBLP:conf\/nips\/Ge0C0L20}.\nBefore training, the memory bank is initialized with the embeddings extracted from the whole training set by using ImageNet-pretrained ResNet-50.\nAt the beginning of each epoch, the k-reciprocal-coded Jaccard distance matrix of all pedestrian features in the bank is used for DBSCAN clustering to obtain pseudo labels $\\mathcal{Y}=\\{y_1,y_2\u2026y_N\\}$.\nBase on the pseudo labels, our method can calculate the common InfoNCE loss~\\cite{DBLP:journals\/corr\/abs-1807-03748} $\\mathcal{L}_{base}$ and our proposed camera-aware contrastive center loss $\\mathcal{L}_{cacc}$.\nLet $\\mathcal{M}$ ($\\mathcal{M} \\in \\mathbb{R}^{d\\times N}$) denotes the memory bank, where $d$ is the dimension of the embeddings.\nAt the end of each training iteration, extracted embeddings $f_i$ are used to update the memory bank $\\mathcal{M}$ at the instance level with the momentum mechanism:\n\n\\vspace{-20pt}\n\\begin{eqnarray}\n  \\mathcal{M}[i]\\leftarrow \\mu \\cdot \\mathcal{M}[i]+\\left(1- \\mu \\right) \\cdot f_{i}~,\n\\end{eqnarray}\n\\vspace{-19pt}\n\n\\noindent where $\\mathcal{M}[i]$ is the $i$-th item in the memory bank, storing the updated feature of sample $x_i$, and $\\mu$ is a momentum coefficient that controls the updating speed.\n\nIn the training mode, the overall objective function of this CA-UReID method can be defined as follows:\n\n\\vspace{-20pt}\n\\begin{eqnarray}\n  \\mathcal{L}_{total}=\\mathcal{L}_{base}+\\lambda_{1} \\cdot \\mathcal{L}_{sep}+\\lambda_{2} \\cdot \\mathcal{L}_{cacc}~,\n  \\label{eq_total_loss}\n\\end{eqnarray}\n\\vspace{-19pt}\n\n\\noindent  where $\\lambda_{1}$ and $\\lambda_{2}$ are hyper-parameters to balance these different components.\nIn the test mode, the $f_{i}$ extracted from image is the embedding used for final matching.\n\n\n\n\n\\vspace{-10pt}\n\\subsection{Camera Style Separation Module} \\label{subsec_CSS}\n\\vspace{-5pt}\n\nThe camera style variance, including camera property and environment information, leads to the gap in intra-class features. \nTo solve this problem, we propose the camera style separation (CSS) module to eliminate the camera style directly from the extracted feature maps.\nInspired by the way DAAM~\\cite{DBLP:conf\/aaai\/HuangPJLX20} separates domain-related information in UDA framework, we design a novel strategy to disentangle camera-style information in our USL pipeline. \nAs shown in Fig.~\\ref{fig_overall_structure}a, the designed CSS module has two branches: the camera-specific branch and the camera-agnostic branch.\n\nIn the camera-specific branch, to separate camera-style features unrelated to pedestrians (e.g. background and camera property, etc.) from feature map, we reframe the learning of this branch as a camera classification problem.\nNote that, although we don't have ID annotation in USL mode, we can use the cameras annotations, which is easy to obtain in the actual collection of datasets.\nSpecifically, $F_{x_i}$ is fed into a Cam-Sep attention module to obtain a weight mask $Attn(F_{x_i})$ to focus on camera information. \nHere, we adopt the HA structure of HA-CNN~\\cite{DBLP:conf\/cvpr\/LiZG18} as the attention module to obtain both spatial and channel attention weights.\nThrough element wise product with the feature map $F_{x_i}$, camera-specific feature map $F_{x_i}^{sp}$ is calculated by the following formula:\n\n\\vspace{-18pt}\n\\begin{eqnarray}\n  F_{x_i}^{sp}=Attn(F_{x_i})\\otimes F_{x_i}~.\n\\end{eqnarray}\n\\vspace{-15pt}\n\n\\noindent The camera-specific embedding $f_i^{sp}$ is generated by GAP and BN operation on $F_{x_i}^{sp}$.\nTo guide the attention of this branch, we design the camera separation loss $\\mathcal{L}_{sep}$:\n\n\\vspace{-18pt}\n\\begin{eqnarray}\n  \\mathcal{L}_{sep}=\\mathbb{E}[-y_i^{cam}log(\\sigma(FC(f_i^{sp})))]~,\n\\end{eqnarray}\n\\vspace{-15pt}\n\n\\noindent where $y_i^{cam}$ is the camera label, $\\sigma$ is the Softmax function, and $FC$ is the fully connected layer.\n$f_i^{sp}$ is input into a classifier with the number of categories as the number of cameras to predict probability of belonging to each camera, which can enforce this branch to extract camera-related information.\n\nIn the camera-agnostic branch, we carry out element-wise products with the complementary mask of $Attn(F_{x_i})$ and \nthe original feature map $F_{x_i}$ to obtain the camera-agnostic feature map $F_{x_i}^{ag}$, which is defined as follows:\n\n\\vspace{-18pt}\n\\begin{eqnarray}\n  F_{x_i}^{ag}=(1-Attn(F_{x_i}))\\otimes F_{x_i}~.\n\\end{eqnarray}\n\\vspace{-15pt}\n\n\\noindent Then the camera-agnostic feature vector $f_i$ is obtained from $F_{x_i}^{ag}$ and used to calculate $\\mathcal{L}_{base}$ and $\\mathcal{L}_{cacc}$.\nBy optimizing $\\mathcal{L}_{sep}$, the influence of camera styles is reduced and the model can learn more robust pedestrian features.\n\n\\vspace{-10pt}\n\\subsection{Camera-aware Contrastive Center Loss} \\label{subsec_CACC}\n\\vspace{-5pt}\n\nTo further enhance the network to extract camera-invariant embeddings $f_i$, we propose a novel camera-aware contrastive center (CACC) loss $\\mathcal{L}_{cacc}$ to narrow the feature distance of different cameras in the same class and increase the feature distance of different classes.\n\nAs shown in Fig.~\\ref{fig_overall_structure}b, different from the common contrastive learning loss $\\mathcal{L}_{base}$ in UReID, the $\\mathcal{L}_{cacc}$ considers the camera in the clustering process.\nBesides, unlike other camera-aware contrastive losses (ICE~\\cite{DBLP:journals\/corr\/abs-2103-16364}, CAP~\\cite{DBLP:conf\/aaai\/0001LHG021}), our proposed $\\mathcal{L}_{cacc}$ constrains the embeddings at the set level by using a camera center instead of a sample as an anchor (Fig.~\\ref{fig_cacc_loss}), attempting to avoid the impact of noise samples.\nSpecifically, given the camera number $c$ and identity class $k$ for each embedding stored in the memory bank $\\mathcal{M}$ and a batch, the camera center $g_k^{c}$ and $p_k^{c}$ can be respectively formulated as:\n\n\\vspace{-13pt}\n\\begin{eqnarray}\n  g^{c}_k=\\frac{1}{N^c_{k}}\\sum_{\\mathcal{M}[i] \\in k \\cap c} \\mathcal{M}[i],~~~~p^{c}_k=\\frac{1}{\\hat{N}^c_{k}}\\sum_{f_{i} \\in k \\cap c} f_{i}~,\n\\end{eqnarray}\n\\vspace{-10pt}\n\n\\noindent where $N^c_{k}$ is the number of all embeddings of identity class $k$ with camera number $c$ in $\\mathcal{M}$, and $\\hat{N}^c_{k}$ is the number of embeddings with class $k$ and camera number $c$ in a training batch.\n\nIn each batch, by using $p^{c}_k$ as an anchor and  \ncombining all positive camera centers $g_i$ and k-nearest negative camera centers $g_j$, \nour proposed $\\mathcal{L}_{cacc}$ can be formulated as:\n\n\\vspace{-13pt}\n\\begin{eqnarray}\n  \\mathcal{L}_{cacc}=-\\frac{1}{G}\\sum_{i\\in k}log \\frac{S(p^{c}_k, g_i)}{S(p_k^{c} , g_i)+\\sum_{j=1}^{J}S(p_k^{c} , g_j)}~,\n\\end{eqnarray}\n\\vspace{-10pt}\n\n\\noindent where $G$ is the number of $g_i$, $J$ is the number of $g_j$, $S(p_k^{c} , g_i)=exp(p_k^{c} \\cdot g_i\/\\tau_{cacc})$, and $\\tau_{cacc}$ is a temperature hyper-parameter.\nBy optimizing the $\\mathcal{L}_{cacc}$ constrained on the center within each camera (shown in Fig.~\\ref{fig_cacc_loss}), our method can effectively pull the intra-class features and push the inter-class features to obtain more discriminative embeddings.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{camera-aware_contrastive_center_loss2.pdf}\n\\vspace{-18pt}\n\\caption{The camera-aware contrastive center loss. The geometrical patterns with different colors represent different ID's embeddings, and different shapes represent different cameras.}\n\\vspace{-10pt}\n\\label{fig_cacc_loss}\n\\end{figure}\n\n\n\\vspace{-5pt}\n\\section{Experiments}\n\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Comparison with state-of-the-art methods.}\n  \\fontsize{8}{9.5}\\selectfont\n  \\setlength{\\tabcolsep}{0.4mm}\n  \\begin{tabular}{c|c|c|cc|c|cc}\n  \\hline\n  \\hline\n  \\multirow{2}[0]{*}{Method} & \\multirow{2}[0]{*}{Venue} & \\multirow{2}[0]{*}{Source} & \\multicolumn{2}{c|}{Market} & \\multirow{2}[0]{*}{Source} & \\multicolumn{2}{c}{Duke} \\\\\\cline{4-5}\\cline{7-8}\n        &       &       & mAP   & Rank1  &       & mAP   & Rank1  \\\\\n  \\hline\n  \\multicolumn{8}{l}{\\textbf{Unsupervised Domain Adaptation}}    \\\\\n  \\hline\n \n  MMCL~\\cite{DBLP:conf\/cvpr\/WangZ20a}       & CVPR2020 & Duke  &  60.4 &  84.4  & Market &  51.4 & 72.4  \\\\\n \n  DAAM~\\cite{DBLP:conf\/aaai\/HuangPJLX20}    & AAAI2020 & Duke  & 67.8  & 86.4   & Market & 63.9  & 77.6  \\\\\n  HGA~\\cite{DBLP:conf\/aaai\/ZhangLLGDLJ21}   & AAAI2021 & Duke  & 70.3  & 89.5   & Market & 67.1  & 80.4  \\\\\n  MMT~\\cite{DBLP:conf\/iclr\/GeCL20}          & ICLR2020 & Duke  & 71.2  & 87.7   & Market & 65.1  & 78.0  \\\\\n  JVTC+*~\\cite{DBLP:conf\/cvpr\/ChenWLDB21}   & CVPR2021 & Duke  & 75.4  & 90.5   & Market & 67.6  & 81.9  \\\\\n  MetaCam~\\cite{DBLP:conf\/cvpr\/YangZLCLLS21}&CVPR2021  & Duke  & 76.5  & 90.1   & Market & 65.0  & 79.5  \\\\\n  SpCL~\\cite{DBLP:conf\/nips\/Ge0C0L20}       &NeurIPS2020& Duke & 76.7  & 90.3   & Market & 68.8  & 82.9  \\\\\n  GCMT~\\cite{DBLP:conf\/ijcai\/LiuZ21}        & IJCAI2021& Duke  & 77.1  & 90.6   & Market & 67.8  & 81.1  \\\\\n  UNRN~\\cite{DBLP:conf\/aaai\/ZhengLZZZ21}    & AAAI2021 & Duke  & 78.1  & 91.9   & Market & 69.1  & 82.0  \\\\\n  GLT~\\cite{DBLP:conf\/cvpr\/ZhengLH0LZ21}    & CVPR2021 & Duke  & 79.5  & 92.2   & Market & 69.2  & 82.0  \\\\\n  OPLG~\\cite{Zheng_2021_ICCV}               & ICCV2021 & Duke  & 80.0  & 91.5   & Market & 70.1  & 82.2  \\\\\n  \\hline\n  \\multicolumn{8}{l}{\\textbf{Purely Unsupervised Learning}}    \\\\\n  \\hline\n  BUC~\\cite{DBLP:conf\/aaai\/LinD00019}  & AAAI2019 &  None &  38.3 & 66.2   & None  & 27.5  & 47.4   \\\\\n  SSL~\\cite{DBLP:conf\/cvpr\/LinXWY020}  & CVPR2020 &  None &  37.8 & 71.7   & None  & 28.6  & 52.5   \\\\\n  MMCL~\\cite{DBLP:conf\/cvpr\/WangZ20a}  & CVPR2020 &  None &  45.5 & 80.3   & None  & 40.2  & 65.2   \\\\\n  HCT~\\cite{DBLP:conf\/cvpr\/ZengNW020}  & CVPR2020 &  None &  56.4 & 80.0   & None  & 50.7  & 69.6   \\\\\n \n  SpCL~\\cite{DBLP:conf\/nips\/Ge0C0L20}  & NeurIPS2020 &None&  73.1 & 88.1   & None  & 65.3  & 81.2   \\\\\n  IICS~\\cite{DBLP:conf\/cvpr\/XuanZ21}   & CVPR2021 & None  & 72.1  & 88.8   & None  & 59.1  & 76.9 \\\\\n  OPLG~\\cite{Zheng_2021_ICCV}          & ICCV2021 & None  & 78.1  & 91.1   & None  & 65.6  & 79.8 \\\\\n  CAP~\\cite{DBLP:conf\/aaai\/0001LHG021} & AAAI2021 & None  & 79.2  & 91.4   & None  & 67.3  & 81.1 \\\\\n  MGH~\\cite{DBLP:conf\/mm\/WuWLT21}      & MM2021   & None  & 81.7  & 93.2   & None  & 70.2  & 83.7 \\\\\n \n \n  ICE~\\cite{DBLP:journals\/corr\/abs-2103-16364}   & ICCV2021 & None  & 82.3  & 93.8 & None  & 69.9  & 83.3 \\\\\n  \\hline\n  \\textbf{CA-UReID (Ours)}  &   -    & \\textbf{None}  & \\textbf{84.5} & \\textbf{94.1}  & \\textbf{None}  & \\textbf{73.2} & \\textbf{85.2} \\\\\n  \\hline\n  \\hline\n  \\end{tabular}\n  \\label{tab_sota_comparison}\n  \\vspace{-10pt}\n\\end{table}\n\n\\vspace{-5pt}\n\\subsection{Datasets and Implementation Details}\n\\vspace{-5pt}\n\n\\textbf{Datasets.} We use the mainstream ReID datasets to evaluate our proposed method: \nMarket1501~\\cite{DBLP:conf\/iccv\/ZhengSTWWT15} contains 32,668 images of 1,501 identities captured by 6 cameras, while DukeMTMC-reID~\\cite{DBLP:conf\/eccv\/RistaniSZCT16} contains 36,411 images of 1,404 identities captured by 8 cameras.\nAs the common practice, Cumulative Matching Characteristic (CMC) and mean Average Precision (mAP) are used to measure the performance. \n\n\\noindent \\textbf{Implementation Details.}\nWe use ResNet-50 as backbone with the pre-trained weights on ImageNet. \nAll images are resized to $256\\times128$.\nThe data augmentation strategies used in training include random flipping, random cropping, and random erasing.\nIn each epoch, our method uses DBSCAN clustering with k-reciprocal Jaccard distance to generate pseudo-labels. \nThe $eps$ in DBSCAN is set to 0.5\/0.6 for Market1501\/DukeMTMC-reID. \nMomentum factor $\\mu$ is set to 0.2\/0.1 on Market1501\/DukeMTMC-reID.\nIn $\\mathcal{L}_{cacc}$, the temperature factor $\\tau_{cacc}$ is set to 0.07, and $J$ is set to 50.\nThe $\\lambda_{1}$ and $\\lambda_{2}$ in $\\mathcal{L}_{total}$ is empirically set to 0.4 and 1.0,\nThe batch size is set to 64 with Adam optimizer. \nThe learning rate is set to 0.00035 and multiplied by 0.1 per 20 epoch in training.\n\n\\vspace{-10pt}\n\\subsection{Comparison with the State-of-the-art}\n\\vspace{-5pt}\n\nIn experiments, we compare our method with state-of-the-art unsupervised ReID methods, including the UDA and USL methods.\nThe experimental results are shown in Tab.~\\ref{tab_sota_comparison}.\n\nWe list state-of-the-art UDA methods in the top half of Tab.~\\ref{tab_sota_comparison}.\nThe UDA leverages the annotation information of the source domain to make it easier to achieve better performance on the target dataset. \nNevertheless, without any identity annotation, the performance of our method still clearly surpasses them, outperforming the latest OPLG method with +4.5\\% mAP and +2.6\\% Rank-1 on Market1501 and +3.1\\% mAP and +3.0\\% Rank-1 on DukeMTMC-reID.\nAs shown in the bottom half of Tab.~\\ref{tab_sota_comparison}, our method also achieves better performance than SOTA methods under the purely unsupervised setting.\nCompared with IICS~\\cite{DBLP:conf\/cvpr\/XuanZ21}, which also considers the influence of camera style, our method has obvious advantages. \nIICS divides the training process into the intra-camera training stage and the inter-camera training stage, requiring additional AIBN to integrate the knowledge learned under different cameras and reduce the intra-camera variance. \nOur proposed camera separation structure can separate the feature map, which is more direct than IICS.\nCompared with CAP~\\cite{DBLP:conf\/aaai\/0001LHG021} and ICE~\\cite{DBLP:journals\/corr\/abs-2103-16364} using camera-aware proxys, our method still achieve better performance on both Market1501 and DukeMTMC-reID. \nIt also indicates that our designed center-based cross-camera loss has superiority with less sensitivity to noise samples. \n\n\\vspace{-10pt}\n\\subsection{Ablation Study}\n\\vspace{-5pt}\n\n\\begin{table}[t]\n  \\centering\n  \\caption{Ablation studies of the designed components.}\n  \\fontsize{9}{9.5}\\selectfont\n  \\setlength{\\tabcolsep}{1.5mm}\n  \\begin{tabular}{c|l|cc|cc}\n  \\hline\n  \\hline\n  \\multicolumn{1}{c|}{\\multirow{2}[0]{*}{Index}} & \\multicolumn{1}{c|}{\\multirow{2}[0]{*}{Methods}} & \\multicolumn{2}{c|}{Market1501} & \\multicolumn{2}{c}{Duke-reID} \\\\\\cline{3-6}\n      &  & mAP   & Rank1    & mAP   & Rank1 \\\\\n  \\hline\n  1 & Baseline & 78.7  & 91.3  & 68.6  & 82.2 \\\\\n  2 & Baseline~+~CSS & 80.7  & 91.9  & 69.5  & 82.4 \\\\\n  3 & Baseline~+~$\\mathcal{L}_{cacc}$ & 82.8  & 93.3  & 71.9  & 85.0 \\\\\n  4 & Baseline~+~CSS~+~$\\mathcal{L}_{cacc}$ & \\textbf{84.5}  & \\textbf{94.1}  & \\textbf{73.2}  & \\textbf{85.2} \\\\\n  \\hline\n  5 & $\\mathcal{L}_{casc}$ (with sample) & 83.5  & 93.4  & 72.0  & 84.0 \\\\\n  6 & $\\mathcal{L}_{cacc}$ (with center) & \\textbf{84.5}  & \\textbf{94.1}  & \\textbf{73.2}  & \\textbf{85.2} \\\\\n  \\hline\n  \\hline\n  \\end{tabular}\n  \\label{tab_ablation}\n  \\vspace{-5pt}\n\\end{table}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{ccs_attention_map6.pdf}\n  \\vspace{-22pt}\n  \\caption{Visualization of attention map in the CSS module.}\n  \\vspace{-15pt}\n  \\label{fig_css_attention}\n\\end{figure}\n\nIn this section, we further discuss and analyze the effectiveness of key components, including the camera style separation module and the camera-aware contrastive center loss.\n\nAs shown in Tab.~\\ref{tab_ablation}, the Baseline (Index-1) is the method without our proposed CSS branch and CACC loss.\nCompared with the Baseline, the CSS module brings performance improvements on all two metrics (mAP, Rank1) are 2.0\\%, 0.6\\% on Market1501 and 0.9\\%, 0.2\\% on DukeMTMC-reID. \nThe improvements of $\\mathcal{L}_{cacc}$ (Index-3) over baseline are 4.1\\%, 2.0\\% on Market and 3.3\\%, 2.8\\% on Duke.\nThe best performance can be obtained by using both CSS and $\\mathcal{L}_{cacc}$ (Index-4), with improvements of 5.8\\%, 2.8\\% on Market and 4.6\\%, 3.0\\% on Duke. \nIt shows the good complementarity of CSS and $\\mathcal{L}_{cacc}$ and they can jointly optimize the feature learning.\n\nTo observe the function of CSS module, we visualize the learned attention maps in the HA structure.\nAs shown in Fig.~\\ref{fig_css_attention}, the camera-specific maps mainly focus on background, which is more related to cameras because of their fixed position.\nIn contrast, the complementary camera-agnostic attention maps focus on the main body of pedestrian, which can narrow the gap between cameras within the class.\nIn experiments, we also compare two different constraints: the camera-aware contrastive sample loss $\\mathcal{L}_{casc}$ and camera-aware contrastive center loss $\\mathcal{L}_{cacc}$, which act on samples and centers, respectively.\nFrom Tab.~\\ref{tab_ablation}, we can notice that the version of center-based loss achieves higher performance with stronger retrieval ability.\nIn general, by using two components together, our method can generate more discriminative embeddings in this person ReID task under USL settings.\n\n\n\\vspace{-10pt}\n\\section{Conclusion}\n\\vspace{-5pt}\n\nIn this paper, we propose a camera-aware style separation and contrastive learning method for purely unsupervised person ReID task, which can effectively separate camera style from the feature map by our proposed camera style separation module and camera-aware contrastive center loss. \nExperimental results demonstrate that our method shows superior performance against the existing state-of-the-art methods.\n\n\n\\clearpage\n\\bibliographystyle{IEEEbib}\n\\begin{spacing}{0.9}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}