questionQuestion exampleExample remarkRemark

#  On the typical and atypical solutions to the Kuramoto equations††thanks:
Submitted to the editors DATE. TC and EK are supported by a grant from the
Auburn University at Montgomery Research Grant-in-Aid Program and the National
Science Foundation under Grant No. 1923099. EK is also supported by the
Undergraduate Research Experience program funded by the Department of
Mathematics at Auburn University at Montgomery.

Tianran Chen Department of Mathematics, Auburn University at Montgomery,
Montgomery, AL<EMAIL_ADDRESS>Evgeniia Korchevskaia School of Mathematics,
Georgia Institute of Technology  Julia Lindberg Max Planck Institute for
Mathematics in the Sciences, Leipzig, Germany<EMAIL_ADDRESS>
###### Abstract

The Kuramoto model is a dynamical system that models the interaction of
coupled oscillators. There has been much work to effectively bound the number
of equilibria to the Kuramoto model for a given network. By formulating the
Kuramoto equations as a system of algebraic equations, we first relate the
complex root count of the Kuramoto equations to the combinatorics of the
underlying network by showing that the complex root count is generically equal
to the normalized volume of the corresponding adjacency polytope of the
network. We then give explicit algebraic conditions under which this bound is
strict and show that there are conditions where the Kuramoto equations have
infinitely many equilibria.

###### keywords:

Kuramoto model, adjacency polytope, Bernshtein-Kushnirenko-Khovanskii bound

14Q99, 65H10, 52B20

##  1 Introduction

The Kuramoto model [24] is a simple mathematical model that describes the
dynamics on networks of oscillators. It has found applications in
neuroscience, biology, chemistry and power systems [4, 18, 20, 31]. Despite
its simplicity, it exhibits interesting emergent behaviors. Of particular
interest is the phenomenon of frequency synchronization which is when
oscillators spontaneously synchronize to a common frequency. In a rotational
frame, a synchronization configuration is an equilibrium of the dynamical
system, i.e., a root of the Kuramoto equations. This paper aims to understand
the structure of these roots in “typical” and “atypical” networks.

Earlier studies focused on statistical analysis of infinite networks [24] but
more recently, tools from differential and algebraic geometry have enabled
analysis of synchronization on finite networks. For a finite network, knowing
the total number of synchronization configurations is fundamental to our
understanding of this model. From a computational perspective, this knowledge
also plays a critical role in developing numerical methods for finding
synchronization configurations. For instance, this number serves as a stopping
criterion for _monodromy_ algorithms [27] and allows for the development of
specialized _homotopy_ algorithms [9, 11] for finding synchronization
configurations.

In 1982, Ballieul and Byrnes introduced root counting techniques from
algebraic geometry to this field and showed that a Kuramoto network of $N$
oscillators has at most $\binom{2N-2}{N-1}$ synchronization configurations
[2]. Algebraic geometers will recognize this bound as the bi-homogeneous
Bézout bound for an algebraic version of the defining equations. This upper
bound can be reached when the network is complete and complex roots (i.e.,
complex relaxation of the configurations) are counted. However, for sparse
networks, the root count (even counting complex roots) can be significantly
lower than this upper bound [19, 29], demonstrating the need for a network-
dependent root count.

Guo and Salam initiated one of the first algebraic analyses on such sparsity-
dependent root counts [19]. Molzahn, Mehta, and Niemerg provided computational
evidence for the connection between this root count and network topology [29].
In the special case of rank-one coupling, Coss, Hauenstein, Hong and Molzahn
proved this complex root count to be $2^{N}-2$, which is also an
asymptotically sharp bound on the real root count [16]. Chen, Davis and Mehta
established the maximum complex root count for cycle networks to be
$N\binom{N-1}{\lfloor(N-1)/2\rfloor}$ [13], which is sharp as it can be
reached with generic choices of coefficients, and it is asymptotically smaller
than the bi-homogeneous Bézout bound discovered by Baillieul and Byrnes,
proving that sparse networks have significantly fewer synchronization
configurations than dense ones. Interestingly, Lindberg, Zachariah, Boston,
and Lesieutre showed that this bound is attainable by real roots [28]. This
bound is an instance of the “adjacency polytope bound” [10]. However, the
following was still unknown.

######  Question 1.1.

For generic choices of coupling coefficients and natural frequencies, will the
complex root count for the algebraic Kuramoto equations reach the adjacency
polytope bound for all networks? If true, this system is Bernshtein-general,
despite the constraints on the coefficients.

In the first part of this paper, we provide a positive answer to this question
and thus establish the generic root count for the Kuramoto equations derived
from a graph $G$ to be the normalized volume of the adjacency polytope
$\operatorname{conv}\\{\vec{e}_{i}-\vec{e}_{j}\\}_{\\{i,j\\}\in\mathcal{E}(G)}$.
We note that this result is similar to recent work in [5] which shows that the
number of (approximate) complex solutions to the Duffing equations is
generically the volume of the _Oscillator polytope_. We also extend this
generic root count result to variations of the Kuramoto equations.

Even though the generic behavior holds for almost all parameters,
understanding the two types of non-generic behavior is still of interest.
First, the roots of the Kuramoto equations may remain isolated but the total
number can drop below the generic root count. The second part of this paper
considers this situation and aims to answer the following.

######  Question 1.2.

What are the conditions on the coupling coefficients under which the complex
roots count for Kuramoto equations will drop below the generic root count?

Furthermore, for certain networks, we answer the following.

######  Question 1.3.

What is the gap between the generic root count and actual root count?

For the second type of non-generic behavior, non-isolated solutions may
appear. Ashwin, Bick, and Burylko analyzed certain types of non-isolated
solutions in complete networks of identical oscillators [1]. A concrete
example of this in a network of four identical oscillators with uniform
coupling coefficients was described in [16, Example 2.1]. The existence of
non-isolated equilibria for cycle networks was discovered by Lindberg,
Zachariah, Boston and Lesieutre [28]. Recent work by Sclosa shows that for
every $d\geq 1$ there is a Kuramoto network whose stable equilibria form a
manifold of dimension $d$ [32]. In the last part, we aim to answer the
following.

######  Question 1.4.

What are the conditions on the coupling coefficients under which there exist
non-isolated synchronization configurations? What are these configurations?

The rest of this paper is structured as follows. In Section 2, we review
concepts that will be used throughout this paper. We then consider the generic
root count of the Kuramoto equations in Section 3 and provide an answer to
Question 1.1. Next, we turn our attention to non-generic coupling
coefficients. In Section 4 we consider when all solutions are isolated and the
number of complex solutions drops, answering Questions 1.2 and 1.3. In Section
5 we identify parameter values under which the Kuramoto equations have
positive dimensional components, answering Question 1.4. Finally, we conclude
with a few remarks in Section 6.

##  2 Notation and preliminaries

Column vectors, representing points of the lattice $L\cong\mathbb{Z}^{n}$, are
denoted by lowercase letters with arrowhead, e.g., $\vec{a}$. We use boldface
letters, e.g., $\boldsymbol{x}$, for points in $\mathbb{C}^{n}$,
$\mathbb{R}^{n}$, or the dual lattice $L^{\vee}\cong\mathbb{Z}^{n}$, and they
are written as row vectors. For $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and
$\vec{a}=(a_{1},\dots,a_{n})^{\top}\in\mathbb{Z}^{n}$,
$\boldsymbol{x}^{\vec{a}}=x_{1}^{a_{1}}\,\cdots\,x_{n}^{a_{n}}$ is a _Laurent
monomial_ in $\boldsymbol{x}$ with the convention that $x^{0}=1$ (including
$x=0$). A _Laurent polynomial_ , $f$, is a linear combination of Laurent
monomials and its _support_ and _Newton polytope_ are denoted
$\operatorname{supp}(f)$ and $\operatorname{Newt}(f)$, respectively. With
respect to a vector $\boldsymbol{v}$, the _initial form_ of $f$ is the Laurent
polynomial
$\operatorname{init}_{\boldsymbol{v}}(f)(\boldsymbol{x}):=\sum_{\vec{a}\in(S)_{\vec{v}}}c_{\vec{a}}\,\boldsymbol{x}^{\vec{a}}$,
where $(S)_{\vec{v}}$ is the subset of $S$ on which
$\left\langle\,\boldsymbol{v}\,,\,\cdot\,\right\rangle$ is minimized. For an
integer matrix $A=[\vec{a}_{1}\;\cdots\;\vec{a}_{m}]$,
$\boldsymbol{x}^{A}:=(\boldsymbol{x}^{\vec{a}_{1}},\dots,\boldsymbol{x}^{\vec{a}_{m}})$
is a system of Laurent monomials, and we restrict the domain to the _algebraic
torus_ $(\mathbb{C}^{*})^{n}=(\mathbb{C}\setminus\\{0\\})^{n}$ whose group
structure is given by
$(x_{1},\dots,x_{n})\circ(y_{1},\dots,y_{n}):=(x_{1}y_{1},\dots,x_{n}y_{n})$.

We fix $G$ to be a connected graph with vertex set
$\mathcal{V}(G)=\\{0,1,\dots,n\\}$ and edge set
$\mathcal{E}(G)\neq\varnothing$. For node $i$ of $G$, $\mathcal{N}_{G}(i)$ is
the set of its adjacent nodes. Arrowheads will be used to distinguish digraphs
from graphs, e.g., $\vec{G}$. A more thorough list of notation is included in
Appendix A.

###  2.1 The Kuramoto model

A network of $n+1$ coupled oscillators can be modeled by a graph $G$ with
nodes $\mathcal{V}(G)=\\{0,\dots,n\\}$ and edges $\mathcal{E}(G)$ representing
the oscillators and their connections, respectively. Oscillators have natural
frequencies $\vec{w}=(w_{0},w_{1},\ldots,w_{n})^{\top}$, and, along the edges
in $G$, nonzero constants $K=\\{k_{ij}\\}$ with $k_{ij}=k_{ji}$ quantify the
coupling strength. The structure $(G,K,\vec{w})$ encoding this model will
simply be called a network. The dynamics are described by

(1)
$\frac{d\theta_{i}}{dt}=w_{i}-\sum_{j\in\mathcal{N}_{G}(i)}k_{ij}\sin(\theta_{i}-\theta_{j})\quad\text{for
}i=0,\dots,n,$

where $\theta_{i}$ is the phase angle of the $i$-th oscillator [24].
_Frequency synchronization configurations_ are defined to be configurations of
$(\theta_{0},\dots,\theta_{n})$ for which all oscillators are tuned to have
the exact same angular velocity — the average of the natural frequencies
$\overline{w}$. By adopting a rotational frame, we can assume $\theta_{0}=0$.
Since $k_{ij}=k_{ji}$, we can also eliminate one equation. Therefore, the
(frequency) synchronization configurations are the zeroes to the system of $n$
transcendental functions:

(2)
$(w_{i}-\overline{w})-\sum_{j\in\mathcal{N}_{G}(i)}k_{ij}\sin(\theta_{i}-\theta_{j})\quad\text{for
}i=1,\dots,n.$

The central question of finding the maximum number of synchronization
configurations of a Kuramoto network is therefore equivalent to the root
counting question for this system.

###  2.2 Algebraic Kuramoto equations

To leverage the power of root counting techniques from algebraic geometry, the
above transcendental system can be reformulated into an algebraic system via
the change of variables $x_{i}=e^{\mathfrak{i}\theta_{i}}$ where
$\mathfrak{i}=\sqrt{-1}$. Then
$\sin(\theta_{i}-\theta_{j})=\frac{1}{2\mathfrak{i}}(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}})$,
and (2) is transformed into a system of $n$ Laurent polynomials
$\vec{f}_{G}=(f_{G,1},\dots,f_{G,n})^{\top}$, given by

(3)
$f_{G,i}(x_{1},\dots,x_{n})=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}a_{ij}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right)\quad\text{for
}i=1,\dots,n,$

where $a_{ji}=a_{ij}=\frac{k_{ij}}{2\mathfrak{i}}$, and
$\overline{w}_{i}=w_{i}-\overline{w}$. This system will be referred to as the
algebraic Kuramoto system,111$f_{G}$ depends on the choice of coupling
coefficients $K=[k_{ij}]$ and the natural frequencies $\vec{w}$, and we will
use the notation $f_{(G,K)}$ or $f_{(G,K,\vec{w})}$ to emphasize these
dependencies, when needed.  and it captures all synchronization configurations
in the sense that the real zeros to (2) correspond to the complex zeros of (3)
on the real torus $(S^{1})^{n}$ (i.e., $|x_{i}|=|e^{\mathfrak{i}\theta}|=1$).

In much of this paper, we relax the root-counting problem by considering all
$\mathbb{C}^{*}$-zeros of (3).222 Throughout this paper, we talk about
$\mathbb{C}^{*}$-zeros of (3), even though there is no difference between
$\mathbb{C}^{*}$-zeros and $\mathbb{C}$-zeros. We emphasize this distinction
because in recent algebraic studies of Kuramoto equations, other algebraic
formulations of (2) have been used, and the phrase “complex zeros/roots of the
Kuramoto equations” has a broader meaning, which include zeros at “toric
infinity” in the formulation used in this paper.  One important observation is
that we can ignore pendant nodes $G$ in the sense that synchronization
configurations naturally extend to pendant nodes.

######  Lemma 2.1.

[26, Theorem 2.5.1] Suppose $v\neq 0$ is a pendant node of $G$. Let
$G^{\prime}=G-\\{v\\}$. Then any $\mathbb{C}^{*}$-zero of
$\vec{f}_{G^{\prime}}$ extends to two distinct $\mathbb{C}^{*}$-zeros for
$\vec{f}_{G}$.

###  2.3 Power flow equations

The _PV-type power flow system_ is one important variation of the Kuramoto
system. In it, the graph $G$ models an electric power network where
$\mathcal{V}(G)=\\{0,\ldots,n\\}$ represent buses in the power network. An
edge $\\{i,j\\}\in\mathcal{E}(G)$, representing the connection between buses
$i$ and $j$, has a known complex admittance
$b_{ij}^{\prime}+\mathfrak{i}g_{ij}^{\prime}$. For each bus $i$, the
relationship between its complex power injection $P_{i}+\mathfrak{i}Q_{i}$ and
the complex voltages is captured by the nonlinear equations

(4) $\displaystyle P_{i}$
$\displaystyle=\sum_{j\in\mathcal{N}_{G}(i)}|V_{i}||V_{j}|(g_{ij}^{\prime}\cos(\theta_{i}-\theta_{j})+b_{ij}^{\prime}\sin(\theta_{i}-\theta_{j}))$
(5) $\displaystyle Q_{i}$
$\displaystyle=\sum_{j\in\mathcal{N}_{G}(i)}|V_{i}||V_{j}|(g_{ij}^{\prime}\sin(\theta_{i}-\theta_{j})-b_{ij}^{\prime}\cos(\theta_{i}-\theta_{j}))$

where $V_{i}$ and $\theta_{i}$ are the magnitude and the phase angle of the
complex voltage at bus $i$. We fix bus $0$ to be the _slack bus_ , meaning
$\theta_{0}=0$. Equations (4)-(5) are the _power flow equations_.

The case where $Q_{i}$ and $\theta_{i}$ are unknown while $P_{i}$ and
$|V_{i}|$ are known constants are called PV nodes and they model typical
generator buses. As above, through the change of variables
$x_{i}=e^{\mathfrak{i}\theta_{i}}$ we get the _PV-type algebraic power flow
equations_

(6) $\displaystyle f_{G,i}(x_{1},\ldots,x_{n})$
$\displaystyle=P_{i}-\sum_{j\in\mathcal{N}_{G}(i)}g_{ij}\left(\frac{x_{i}}{x_{j}}+\frac{x_{j}}{x_{i}}\right)+b_{ij}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right),\quad\text{for
}i=1,\ldots,n$

where $b_{ij}=\frac{1}{2\mathfrak{i}}|V_{i}||V_{j}|b_{ij}^{\prime}$ and
$g_{ij}=\frac{1}{2}|V_{i}||V_{j}|g_{ij}^{\prime}$. When $g_{ij}=0$, the
corresponding power system is _lossless_ and (6) reduces to (3). Otherwise,
the system is _lossy_ and (6) differs from (3) in the constraints on the
coefficients. Yet, the same set of monomials are involved, and as we will
demonstrate, the algebraic arguments we will develop can be applied to this
generalization.

###  2.4 Kuramoto equations with phase delays

Another generalization of the Kuramoto model considers _phase delays_. With
known phase delay parameters $\delta_{ij}\in\mathbb{R}$, (2) is augmented to:

$0=w_{i}-\overline{w}-\sum_{j\in\mathcal{N}_{G}(i)}k_{ij}\sin(\theta_{i}-\theta_{j}+\delta_{ij}),\quad\text{for
}i=1,\ldots,n,$

which corresponds to the system in which for a pair of coupled oscillators $i$
and $j$, oscillator $i$ responds not directly to the phase angle of oscillator
$j$ but its delayed phase $\theta_{j}-\delta_{ij}$. Letting
$C_{ij}=e^{{\mathbf{i}}\delta_{ij}}$, we can again make this system algebraic
giving:

(7) $\displaystyle
f_{G,i}(x_{1},\ldots,x_{n})=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}a_{ij}\left(\frac{x_{i}C_{ij}}{x_{j}}-\frac{x_{j}}{x_{i}C_{ij}}\right),\quad\text{for
}i=1,\ldots,n.$

This is, again, a generalization of the algebraic Kuramoto system (3) that
involves the same set of monomials. The algebraic results to be developed also
apply to this family of algebraic systems.

###  2.5 BKK bound

The root counting arguments in this paper revolve around the Bernshtein-
Kushnirenko-Khovanskii (BKK) bound, especially Bernshtein’s Second Theorem.

######  Theorem 2.2 (D. Bernshtein 1975 [3]).

(A) For a square Laurent system $\vec{f}=(f_{1},\dots,f_{n})$, if for all
nonzero vectors $\vec{v}\in\mathbb{R}^{n}$,
$\operatorname{init}_{\vec{v}}(\vec{f})$ has no $\mathbb{C}^{*}$-zeros, then
all $\mathbb{C}^{*}$-zeros of $\vec{f}$ are isolated, and the total number,
counting multiplicity, is the mixed volume
$M=\operatorname{MV}(\operatorname{Newt}(f_{1}),\ldots,\operatorname{Newt}(f_{n}))$.

(B) If $\operatorname{init}_{\vec{v}}(\vec{f})$ has a $\mathbb{C}^{*}$-zero
for some $\vec{v}\neq\vec{0}$, then the number of isolated
$\mathbb{C}^{*}$-zeros $\vec{f}$ has, counting multiplicity, is strictly less
than $M$ if $M>0$.

A system for which the condition (A) holds is said to be _Bernshtein-general_.
Only the special case of identical Newton polytopes, i.e., when
$\operatorname{Newt}(f_{1}),\ldots,\operatorname{Newt}(f_{n})$ are all
identical, will be used. This specialized version strengthens Kushnirenko’s
Theorem [25].

###  2.6 Randomized algebraic Kuramoto system

The analysis of the algebraic Kuramoto system (3) can be further simplified
through “randomization” (a.k.a. “pre-conditioning”). Taking $\vec{f}_{G}$ as a
column vector, for any nonsingular matrix $R$, the systems $\vec{f}_{G}$ and
$\vec{f}^{*}_{G}:=R\,\vec{f}_{G}$ have the same zero set. With a generic
choice of $R$, there will be no complete cancellation of terms, and
$\vec{f}^{*}_{G}$ will be referred to as the randomized (algebraic) Kuramoto
system. This system is _unmixed_ in the sense that for all $i\in[n]$,
$f^{*}_{G,i}$ have identical supports since they involve the same set of
monomials, namely, constant terms, $x_{i}x_{j}^{-1}$ and $x_{j}x_{i}^{-1}$ for
every edge $\\{i,j\\}\in\mathcal{E}(G)$.

The randomization $\vec{f}_{G}\mapsto\vec{f}^{*}_{G}$ does not alter the zero
set but makes a very helpful change to the tropical structure in the sense
that much is already known about the tropical self-intersection of
$\vec{f}^{*}_{G}$. In particular, there is a mapping between the graph-
theoretical features of $G$ and the tropical structures of $\vec{f}^{*}_{G}$
through which we can gain key insight into the structure of the zeros of
$\vec{f}^{*}_{G}$. The next three subsections briefly review the geometric
interpretation of this connection.

###  2.7 Adjacency polytopes

For each $i=1,\ldots,n$, $\operatorname{supp}(f^{*}_{G,i})$ are all identical
and given by

$\check{\nabla}_{G}:=\\{\pm(\vec{e}_{i}-\vec{e}_{j})\mid\\{i,j\\}\in\mathcal{E}(G)\\}\;\cup\;\\{\vec{0}\\},$

where $\vec{e}_{i}$ is the $i$-th standard basis vector of $\mathbb{R}^{n}$
for $i=1,\ldots,n$, and $\vec{e}_{0}=\vec{0}$. In Refs. [10, 13], the convex
hull of $\check{\nabla}_{G}$ is called the _adjacency polytope_ of $G$ (of PV-
type333This is retroactively named the adjacency polytope of PV-type to
distinguish it from those of PQ-type [15]. In general, an adjacency polytope
associated with a graph $G$ is the convex hull of $\\{\vec{0}\\}$ and point
sets $\\{P_{ij}\mid\\{i,j\\}\in\mathcal{E}(G)\\}$ that depend on the “types”
of the nodes, a concept originally derived from the power-flow studies
(Section 2.3). ). It is unimodularly equivalent to the _symmetric edge
polytope_ of $G$ that has appeared in number theory and discrete geometry
(see, for example, the broad overview provided in [17]).

We will not distinguish $\check{\nabla}_{G}$ from its convex hull, meaning a
“face” of $\check{\nabla}_{G}$ refers to a subset
$F\subseteq\check{\nabla}_{G}$ such that $\operatorname{conv}(F)$ is a face of
$\operatorname{conv}(\check{\nabla}_{G})$ and
$\dim(F):=\dim(\operatorname{conv}(F))$. The set of facets and the boundary of
$\check{\nabla}_{G}$ are denoted $\mathcal{F}(\check{\nabla}_{G})$ and
$\partial\check{\nabla}_{G}$, respectively. The _corank_ 444In matroid theory,
the more common term is “ _nullity_ ”. Here, we follow the convention from
convex geometry.  of a face $F$, denoted $\operatorname{corank}(F)$, is the
number $|F|-\dim(F)-1$.

By Kushnirenko’s Theorem [25], the _normalized volume_ of the adjacency
polytope, denoted $\operatorname{Vol}(\check{\nabla}_{G})$, is an upper bound
for the $\mathbb{C}^{*}$-zero count for $\vec{f}^{*}_{G}$ and $\vec{f}_{G}$
which also bounds the real zero count to the transcendental Kuramoto system
(2). This is the _adjacency polytope bound_ [10, 13].

###  2.8 Faces and face subgraphs

There is an intimate connection between faces of $\check{\nabla}_{G}$ and
subgraphs of $G$. Since $\vec{0}$ is an interior point of
$\check{\nabla}_{G}$, every vertex of a proper face $F$ of
$\check{\nabla}_{G}$ is of the form $\vec{e}_{i}-\vec{e}_{j}$ for some
$\\{i,j\\}\in\mathcal{E}(G)$. Thus, it is natural to consider the
corresponding _facial subgraph_ $G_{F}$ and _facial subdigraph_ $\vec{G}_{F}$,
given by

$\displaystyle\mathcal{E}(G_{F})$
$\displaystyle=\\{\\{i,j\\}\mid\vec{e}_{i}-\vec{e}_{j}\in F\text{ or
}\vec{e}_{j}-\vec{e}_{i}\in F\\}\;\text{and}$
$\displaystyle\mathcal{E}(\vec{G}_{F})$
$\displaystyle=\\{(i,j)\mid\vec{e}_{i}-\vec{e}_{j}\in F\\}.$

As defined in [9, 17], for $F\in\mathcal{F}(\check{\nabla}_{G})$,
$\mathcal{E}(G_{F})$ is called a _facet subgraph_ and
$\mathcal{E}(\vec{G}_{F})$ a _facet subdigraph_. Higashitani, Jochemko, and
Michałek provided a topological classification of face subgraphs [22, Theorem
3.1] and it was later reinterpreted [12, Theorem 3]. We state the latter here.

######  Theorem 2.3 (Theorem 3 [12]).

Let $H$ be a nontrivial connected subgraph of $G$.

  1. 1.

$H$ is a face subgraph of $G$ if and only if it is a maximal bipartite
subgraph of $G[\mathcal{V}(H)]$.

  2. 2.

$H$ is a facet subgraph of $G$ if and only if it is a maximal bipartite
subgraph of $G$.

Multiple faces can correspond to the same facial subgraph (in particular,
$G_{F}=G_{(-F)}$ for any face $F$). The crisper parametrization is given by
the correspondence $F\mapsto\vec{G}_{F}$. Ref. [12] describes the balancing
conditions that characterize facial subdigraphs. Here, we only make use of the
balancing condition for the “primitive cycle vectors” associated with a face
subdigraph.

For a facial subdigraph $\vec{G}_{F}$, its _reduced incidence matrix_
$\check{Q}(\vec{G}_{F})$ is the matrix with columns $\vec{e}_{i}-\vec{e}_{j}$
such that $(i,j)\in\mathcal{E}(\vec{G}_{F})$ (see Appendix A). For a face $F$
of $\check{\nabla}_{G}$ such that $G_{F}$ is connected, its corank $d$ equals
the nullity of $\check{Q}(\vec{G}_{F})$. The null space is spanned by $d$
vectors with entries $\\{+1,0,-1\\}$, indicating the incidence of arcs with
the cycles in $G_{F}$ with prescribed orientations. These vectors will be
referred to as _primitive cycle vectors_.

######  Lemma 2.4 (Theorem 7 [12] ).

Let $F$ be a face of $\check{\nabla}_{G}$ for which $G_{F}$ is connected. For
any cycle in $G_{F}$, let $\vec{\eta}$ be its associated primitive cycle
vector with respect to $\vec{G}_{F}$, then
$\left\langle\boldsymbol{1}\,,\,\vec{\eta}\right\rangle=0$.

Note that these conditions are only necessary conditions on the primitive
cycle vectors. If $G$ contains more than one cycle, these conditions are not
sufficient. Stronger classifications results (which are not needed here) can
be found in [12, 22].

For a subdigraph $\vec{H}$ of $\vec{G}$, the _coupling vector_
$\boldsymbol{k}(\vec{H})$ has entries $k_{ij}$ for $(i,j)\in\vec{H}$.
Similarly, the entries of $\boldsymbol{a}(\vec{H})$ are
$a_{ij}=\frac{k_{ij}}{2\mathfrak{i}}$ for $(i,j)\in\mathcal{E}(\vec{H})$. The
ordering of the entries is arbitrary, but when appearing in the same context
with $\check{Q}(\vec{H})$, consistent ordering is implied.

###  2.9 Facial systems

The vast literature on the facial structure of $\check{\nabla}_{G}$ gives us a
shortcut to understanding the initial systems of the randomized algebraic
Kuramoto system $\vec{f}^{*}_{G}$. Recall that
$\operatorname{supp}(f^{*}_{G,i})$ are all identical, so the initial systems
of $\vec{f}^{*}_{G}$ have particularly simple descriptions corresponding to
proper faces of $\check{\nabla}_{G}$. For any
$0\neq\boldsymbol{v}\in\mathbb{R}^{n}$, the initial system
$\operatorname{init}_{\boldsymbol{v}}(\vec{f}^{*}_{G})$ is

(8)
$\operatorname{init}_{\boldsymbol{v}}(f^{*}_{G,i})(\boldsymbol{x})=\sum_{\vec{e}_{j}-\vec{e}_{j^{\prime}}\in
F}c_{i,j,j^{\prime}}\,\boldsymbol{x}^{\vec{e}_{j}-\vec{e}_{j^{\prime}}}=\sum_{(j,j^{\prime})\in\mathcal{E}(\vec{G}_{F})}c_{i,j,j^{\prime}}\,\boldsymbol{x}^{\vec{e}_{j}-\vec{e}_{j^{\prime}}}\quad\text{for
}i=1,\ldots,n,$

where $F$ is the proper face of $\check{\nabla}_{G}$ for which
$\boldsymbol{v}$ is an inner normal vector. We will make frequent use of this
geometric interpretation and therefore it is convenient to slightly abuse the
notation and write
$\operatorname{init}_{F}(\vec{f}^{*}_{G}):=\operatorname{init}_{\vec{v}}(\vec{f}^{*}_{G})$
when the particular choice of vector $\vec{v}$ defining the face $F$ is not
important. It will be called a _facial system_ of $\vec{f}^{*}_{G}$, (or a
_facet system_ if $F\in\mathcal{F}(\check{\nabla}_{G})$).

##  3 Generic complex root count

In this section, we establish the $\mathbb{C}^{*}$-zero count for the
algebraic Kuramoto system (3) for generic choices of real constants
$\overline{w}_{1},\ldots,\overline{w}_{n}$ and
$\\{k_{ij}\\}_{\\{i,j\\}\in\mathcal{E}(G)}$. We show that the generic
$\mathbb{C}^{*}$-zero count is the adjacency polytope bound
$\operatorname{Vol}(\check{\nabla}_{G})$. A corollary is that this system is
Bernshtein-general, despite the constraints on the coefficients.

The first observation is that since $\mathbb{R}$ is Zariski-dense in
$\mathbb{C}$, generic choices of complex parameters can be used without
changing the $\mathbb{C}^{*}$-zero count. That is, it is sufficient to focus
on the root counting problem for the system

$0=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}a_{ij}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right)\qquad\text{for
}i=1,\dots,n,$

in the $n$ unknowns $x_{1},\ldots,x_{n}$ (with $x_{0}=1$) for generic
_complex_ choices555For simplicity, we will assume the constant terms
$\overline{w}_{1},\ldots,\overline{w}_{n}$ are chosen generically. This is
equivalent to choosing $w_{1},\ldots,w_{n}$ generically and leaving $w_{0}$
constrained by these choices. of $\overline{w}_{i},a_{ij}$.

There are three main obstacles. First, in each polynomial the monomials
$x_{i}/x_{j}$ and $x_{j}/x_{i}$ share the same coefficient $a_{ij}$. Second,
for any edge $\\{i,j\\}\in\mathcal{E}(G)$, the $i$-th and $j$-th polynomials
have the terms $a_{ij}(x_{i}x_{j}^{-1}-x_{j}x_{i}^{-1})$ and
$a_{ji}(x_{j}x_{i}^{-1}-x_{i}x_{j}^{-1})$, respectively, which are negations
of each other since $a_{ij}=a_{ji}$. Therefore, the allowed choices of
coefficients consists of a nowhere dense subset of (Lebesgue) measure 0 in the
space of all possible complex coefficients. Finally, unless $G$ is the
complete graph, the Newton polytopes of the algebraic Kuramoto system are not
full-dimensional which prevents simpler arguments (e.g. [8]) from being
applied. We will show that despite this, the maximum $\mathbb{C}^{*}$-zero
count, given by the adjacency polytope bound is generically attained.

Before presenting the main theorem of this section, we first establish a few
technical results that will be used, some of which are well known, but are
nonetheless included for completeness.

######  Lemma 3.1.

The lifting function $\tilde{\omega}:\check{\nabla}_{G}\to\mathbb{Q}$, given
by

$\tilde{\omega}(\vec{x})=\begin{cases}0&\text{if }\vec{x}=\vec{0}\\\
1&\text{otherwise}\end{cases}$

induces a regular subdivision
$\Sigma_{\omega}(\check{\nabla}_{G})=\\{\vec{0}\cup F\mid
F\in\mathcal{F}(\check{\nabla}_{G})\\}$.

This is a well known consequence of assigning a sufficiently small lifting
value to an interior point ($\vec{0}$ in this case). The resulting regular
subdivision may be too coarse to be useful in our discussions. Indeed, it will
not be a triangulation unless $G$ is has no even cycles [12, 11]. It can be
refined into a triangulation through perturbations on the nonzero lifting
values.

######  Lemma 3.2.

For generic but symmetric choices
$\\{\delta_{ij}=\delta_{ji}\in\mathbb{R}\mid\\{i,j\\}\in\mathcal{E}(G)\\}$
that are sufficiently close to 0, the function
$\omega:\check{\nabla}_{G}\to\mathbb{Q}$, given by

$\omega(\vec{a})=\begin{cases}0&\text{if }\vec{a}=\vec{0}\\\
1+\delta_{ij}&\text{if }\vec{a}=\vec{e}_{i}-\vec{e}_{j}\end{cases}$

induces a regular unimodular triangulation
$\Delta_{\omega}=\Sigma_{\omega}(\check{\nabla}_{G})$ that is a refinement of
$\Sigma_{\tilde{w}}(\check{\nabla}_{G})$ and its cells are in one-to-one
correspondence with cells in a unimodular triangulation of
$\partial\check{\nabla}_{G}$. Indeed, each cell is of the form $0\cup\Delta$
where $\Delta$ is a simplex in $\partial\check{\nabla}_{G}$.

######  Proof 3.3.

We first show the interior point $\vec{0}$ is contained in every cell. Let
$C\in\Delta_{\omega}$ and $(\vec{v},1)$ be the inner normal vector the lower
facet of $\check{\nabla}_{G}^{\omega}$ whose projection is $C$. Suppose
$\vec{0}\not\in C$, then $C$ contains an affinely independent set of $n+1$
points $\\{\vec{a}_{0},\ldots,\vec{a}_{n}\\}\not\ni\vec{0}$, and $\vec{v}$
satisfies the equation

$\begin{bmatrix}\vec{a}_{1}^{\top}-\vec{a}_{0}^{\top}\\\ \vdots\\\
\vec{a}_{n}^{\top}-\vec{a}_{0}^{\top}\\\
\end{bmatrix}\,\vec{v}=\begin{bmatrix}\omega(\vec{a}_{0})-\omega(\vec{a}_{1})\\\
\vdots\\\ \omega(\vec{a}_{0})-\omega(\vec{a}_{n})\end{bmatrix}.$

Let $B$ be the matrix on the left and $\vec{\beta}$ be the vector on the
right, then $\vec{v}=B^{-1}\vec{\beta}$ and thus

$|\left\langle\,\vec{a}\,,\,\vec{v}\,\right\rangle|\leq\|\vec{a}\|\|\vec{v}\|\leq\|\vec{a}\|\,\|B^{-1}\|\,\|\vec{\beta}\|\quad\text{for
any }\vec{a}\in C.$

Note that entries of $\vec{\beta}$ are differences among the
$\\{\delta_{ij}\\}$. Therefore, for any $\epsilon>0$, there is a $\delta$ such
that $\delta_{ij}<\delta$ for all $\\{i,j\\}\in\mathcal{E}(G)$ implies
$|\left\langle\,\vec{a}\,,\,\vec{v}\,\right\rangle|<\epsilon$, contradicting
with the assumption that

$\left\langle\,\vec{v}\,,\,\vec{a}\,\right\rangle+\omega(\vec{a})<\langle\,\vec{v}\,,\,\vec{0}\,\rangle+\omega(\vec{0})=0.$

This shows that $\vec{0}$ must be contained in every cell.

To establish $\Delta_{\omega}$ as a triangulation, it is sufficient to show
nonzero points in a cell $C\in\Delta_{\omega}$ are assigned independent
lifting values by $\omega$. If $\pm(\vec{e}_{i}-\vec{e}_{j})\in C$, for some
$\\{i,j\\}\in\mathcal{E}(G)$, then

$\left\langle\,\vec{v}\,,\,\pm(\vec{e}_{i}-\vec{e}_{j})\,\right\rangle+\omega(\pm(\vec{e}_{i}-\vec{e}_{j}))=h,$

where $h$ is the minimum of
$\left\langle\,\vec{v}\,,\,\,\cdot\,\,\right\rangle+\omega(\,\cdot\,)$ over
$\check{\nabla}_{G}$. Since
$\omega(\pm(\vec{e}_{i}-\vec{e}_{j}))=1+\delta_{ij}$, summing the two equation
produces

$1+\delta_{ij}=h>0,$

which contradict with constraint that
$0=\langle\,\vec{v}\,,\,\vec{0}\,\rangle+\omega(\vec{0})\geq h$. Therefore, if
$\vec{e}_{i}-\vec{e}_{j}\in C$, then $\vec{e}_{j}-\vec{e}_{i}\not\in C$.
Consequently, the nonzero points in $C$ are associated with independent
generic choices of lifting values, and thus $C$ must be a simplex.

For unimodularity, note that, since $\vec{e}_{i}-\vec{e}_{j}\in C$ implies
$\vec{e}_{j}-\vec{e}_{i}\not\in C$ for $\\{i,j\\}\in\mathcal{E}(G)$ and for a
cell $C\in\Delta_{\omega}$, the nonzero points of $C$, as vectors, are exactly
columns of (signed) incidence matrix of $G$, which is unimodular. Therefore,
each cell of $\Delta_{\omega}$ is unimodular.

Finally, we show that $\Delta_{\omega}$ is a refinement of
$\Sigma_{\tilde{\omega}}(\check{\nabla})$ from Lemma 3.1. In the following, we
shall fix an arbitrary ordering of the points in $\check{\nabla}_{G}$ and
consider lifting functions on $\check{\nabla}_{G}$, e.g., $\omega$ and
$\tilde{\omega}$, as vectors in $\mathbb{R}^{|\check{\nabla}_{G}|}$, whose
entries are their lifting values. Let
$\mathcal{C}=\mathcal{C}(\check{\nabla}_{G},\Delta_{\omega})$ be the (closed)
secondary cone of $\Delta_{\omega}$ in $\check{\nabla}_{G}$. Then
$\mathcal{C}$ is full-dimensional, since $\Delta_{\omega}$ is a triangulation.
By assumption, $\omega$ is sufficiently close to $\tilde{\omega}$, so
$\tilde{\omega}\in\mathcal{C}$. Consequently,
$\Delta_{\omega}=\Sigma_{\omega}(\check{\nabla}_{G})$ equals or refines
$\Sigma_{\tilde{\omega}}(\check{\nabla}_{G})$.

A lifting function $\omega:\check{\nabla}_{G}\to\mathbb{Q}$ for which Lemma
3.2 holds will be referred to as a generic symmetric lifting function for
$\check{\nabla}_{G}$.

######  Lemma 3.4.

For a generic symmetric lifting function $\omega$ for $\check{\nabla}_{G}$,
let $\Delta$ be a simplex in $\Delta_{\omega}$. Then the digraph
$\vec{G}_{\Delta}$ is acyclic, and its underlying graph $G_{\Delta}$ is a
spanning tree of $G$.

The proof of this lemma is nearly identical to the proof of [9, Theorem 1] and
we include an elementary proof in Appendix B for completeness.

######  Lemma 3.5.

Suppose $\vec{T}<\vec{G}$ is a acyclic, and its underlying graph $T$ is a
spanning tree of $G$, then, for generic
$\overline{w}_{1},\ldots,\overline{w}_{n}\in\mathbb{C}^{*}$ and _any_ choices
of $a_{ij}\in\mathbb{C}^{*}$, the system of $n$ Laurent polynomials

$\overline{w}_{i}-\sum_{j\in\mathcal{N}^{+}_{\vec{T}}(i)}a_{ij}\frac{x_{i}}{x_{j}}+\sum_{j\in\mathcal{N}^{-}_{\vec{T}}(i)}a_{ij}\frac{x_{j}}{x_{i}}\quad\text{for
}i=1,\ldots,n$

has a unique zero in $(\mathbb{C}^{*})^{n}$, and this zero is isolated and
regular.

Here,
$\mathcal{N}^{+}_{\vec{T}}(i)=\\{j\in\mathcal{V}(\vec{T})\mid(i,j)\in\mathcal{E}(\vec{T})\\}$
and
$\mathcal{N}^{-}_{\vec{T}}(i)=\\{j\in\mathcal{V}(\vec{T})\mid(j,i)\in\mathcal{E}(\vec{T})\\}$
are the sets of adjacent nodes of $i$ through outgoing and incoming arcs,
respectively.

######  Proof 3.6.

We prove this by induction on the number of vertices of $G$, $N$. Denote the
above system as $\vec{f}_{\vec{T}}(x_{0},\ldots,x_{N-1})$. The statement is
true for the case where $N=2$ and $|\mathcal{V}(G)|=|\mathcal{V}(T)|=2$.

Assume the statement is true for any graph with $N$ nodes and consider the
case $G$ has $N+1$ nodes $\\{0,1,\ldots,N\\}$. By re-labeling, we can assume
node $N$ is a leaf in $T$ and is adjacent to node $N-1$. Moreover, since this
system is homogeneous of degree 0, we can also scale the homogeneous
coordinates so that $x_{N-1}=1$. Then the above system is decomposed Laurent
system $\vec{f}_{\vec{T}^{\prime}}(x_{0},\ldots,x_{N-1})$ and the binomial
$\overline{w}_{N}\pm a_{N,N-1}x_{N}^{\mp 1}$, where $\vec{T^{\prime}}$ is the
tree $\vec{T}-\\{N\\}$. The induction hypothesis, that
$\vec{f}_{\vec{T^{\prime}}}(x_{0},\ldots,x_{N})$ has a unique isolated and
regular thus completes the proof.

With these technical preparations, we now establish the main theorem of this
section.

######  Theorem 3.7.

For generic choices of real or complex constants
$\overline{w}_{1},\ldots,\overline{w}_{n}$ and generic but symmetric choices
of real or complex coupling coefficients $k_{ij}=k_{ji}\neq 0$ for
$\\{i,j\\}\in\mathcal{E}(G)$, the $\mathbb{C}^{*}$-zero set of the algebraic
Kuramoto system (3) $\vec{f}_{G}=(f_{G,1},\ldots,f_{G,n})$, given by

$f_{G,i}(x_{1},\ldots,x_{n})=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}\frac{k_{ij}}{2\mathfrak{i}}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right)=0$

consists of isolated regular points, and the total number is
$\operatorname{Vol}(\check{\nabla}_{G})$.

This is a far generalization of [13, Corollary 9 and Theorem 16] and [19,
Theorem 3.3.3] the generic $\mathbb{C}^{*}$-zero count is established for
tree666The generic number of complex equilibria for the Kuramoto model on tree
networks is $2^{n}$. This fact appears to be well known among researchers in
power systems long before the referenced paper [13].  cycle, and complete
networks only.

######  Proof 3.8.

By Sard’s Theorem, for generic choices of
$\overline{w}_{1},\ldots,\overline{w}_{n}$, the $\mathbb{C}^{*}$-zero set of
$\vec{f}_{G}$ consists of isolated and regular points. Under this assumption,
we only need to establish the $\mathbb{C}^{*}$-zero count. Moreover, since it
is already known this zero count is bounded by
$\operatorname{Vol}(\check{\nabla}_{G})$, it is sufficient to show that it is
greater than or equal to this bound. We shall take a constructive approach
through a specialized version of the polyhedral homotopy of Huber and
Sturmfels [23].

Let $\omega:\check{\nabla}_{G}\to\mathbb{Q}$ be a generic symmetric lifting
function for $\check{\nabla}_{G}$. We use the notation
$\omega_{ij}=\omega(\vec{e}_{i}-\vec{e}_{j})=\omega(\vec{e}_{j}-\vec{e}_{i})$
and define the function
$\vec{h}=(h_{1},\ldots,h_{n}):(\mathbb{C}^{*})^{n}\times\mathbb{C}\to\mathbb{C}^{n}$,
given by

$h_{i}(x_{1},\ldots,x_{n},t)=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}a_{ij}t^{\omega_{ij}}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right).$

Away from $t=0$, $\vec{h}$ is a parametrized version of the original system
$\vec{f}_{G}$ with coefficients being analytic functions of the parameter $t$.
Note that since $\omega_{ij}=\omega_{ji}$, for any choice of $t\in\mathbb{C}$,
the system still satisfies the symmetry constraints on the coefficients. By
the Parameter Homotopy Theorem [30], for $t$ outside a proper Zariski closed
(i.e., finite) set $Q\subset\mathbb{C}$, the number of isolated
$\mathbb{C}^{*}$-zeros of $\vec{h}(\,\cdot\,,t)=0$ is a constant which is also
an upper bound for the isolated $\mathbb{C}^{*}$-zero count for
$\vec{h}(\,\cdot\,,t)$ for all $t\in\mathbb{C}$. Let $\eta$ be this generic
$\mathbb{C}^{*}$-zero count. Since $\overline{w}_{i}$ and $a_{ij}$ are chosen
generically, and $\vec{h}(\,\cdot\,,1)\equiv\vec{f}_{G}(\;\cdot\;)$, we can
conclude that $t=1$ is outside $Q$ and thus $\eta$ agrees with the generic
$\mathbb{C}^{*}$-zero count of $\vec{f}_{G}$ that we aims to establish. That
is, it is sufficient to show $\eta\geq\operatorname{Vol}(\check{\nabla}_{G})$.

Taking a constructive approach, we now construct $\eta$ smooth curves (of one
real-dimension), that will connect the $\mathbb{C}^{*}$-zeros of $\vec{f}_{G}$
and the collection of $\mathbb{C}^{*}$-zeros of the special systems described
in Lemma 3.5. This connection allows us to count the number of curves.

Along a ray $t:(0,1]\to\mathbb{C}$ on the complex plane parametrized by
$t(\tau)=\tau e^{i\theta}$ for a choice of $\theta\in[0,2\pi)$ that avoids $Q$
(i.e., $t(\tau)\not\in Q$ for all $\tau\in(0,1]$), the function $\vec{h}$ is
given by

$h_{i}(x_{1},\ldots,x_{n},t)=h_{i}(x_{1},\ldots,x_{n},\tau
e^{i\theta})=\overline{w}_{i}-\sum_{j\in\mathcal{N}_{G}(i)}a_{ij}e^{i\theta\omega_{ij}}\tau^{\omega_{ij}}\left(\frac{x_{i}}{x_{j}}-\frac{x_{j}}{x_{i}}\right),$

and $\vec{h}$ has $\eta$ isolated and regular $\mathbb{C}^{*}$-zeros for all
$\tau\in(0,1]$. By the principle of homotopy continuation, the zero set of
$\vec{h}$ form $\eta$ smooth curves in $(\mathbb{C}^{*})^{n}\times(0,1]$
smoothly parametrizable by $\tau$. The problem is now reduced to counting
these curves.

Fixing such a curve $C$, the asymptotic behavior of $C$ as $\tau\to 0$, in a
compactification of $(\mathbb{C}^{*})^{n}$ can be characterized by the
solutions to an initial system of $\vec{h}$, as a system of Laurent
polynomials in $\mathbb{C}\\{\tau\\}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$.
Stated in affine coordinates, by the Fundamental Theorem of Tropical Geometry,
there exists a system of convergent Puiseux series
$x_{1}(\tau),\ldots,x_{n}(\tau)\in\mathbb{C}\\{\tau\\}$ that represents the
germ of $C$, as an analytic variety at $\tau=0$, such that the leading
coefficients, as a point in $(\mathbb{C}^{*})^{n}$ satisfies the initial
system $\operatorname{init}_{\boldsymbol{v}}(\vec{h})$,777We do not claim
$\vec{h}$ form a tropical basis, nor do we require that
$\operatorname{init}_{\boldsymbol{v}}(\vec{h})$ generate the corresponding
initial ideal. Yet, as we will show, an initial system of $\vec{h}$ of the
special form we will describe is sufficient to uniquely determine the leading
coefficients of the Puiseux series expansion of a solution path.  and
$\boldsymbol{v}=(v_{1},\ldots,v_{n})\in\mathbb{Q}^{n}$ are the orders of the
Puiseux series $x_{1}(\tau),\ldots,x_{n}(\tau)\in\mathbb{C}\\{\tau\\}$.
Therefore, it is sufficient to show that there are at least
$\operatorname{Vol}(\check{\nabla}_{G})$ distinct initial systems of $\vec{h}$
that will each contribute one solution path.

By Lemma 3.2, the regular subdivision
$\Delta_{\omega}=\Sigma_{\omega}(\check{\nabla}_{G})$ is a unimodular
triangulation. Fix a simplex $\Delta\in\Delta_{\omega}$, and let
$(\boldsymbol{v},1)$ be the upward pointing inner normal vector that defines
$\Delta$, then by construction, there is an affinely independent set
$\\{\vec{a}_{1},\ldots,\vec{a}_{n}\\}\subset\partial\check{\nabla}_{G}$ such
that

$\displaystyle\left\langle\,\boldsymbol{v}\,,\,\vec{a}_{k}\,\right\rangle+\omega(\vec{a}_{k})$
$\displaystyle=0\quad\text{for }k=1,\ldots,n,\;and$
$\displaystyle\left\langle\,\boldsymbol{v}\,,\,\vec{a}\,\right\rangle+\omega(\vec{a})$
$\displaystyle>0\quad\text{for all
}\vec{a}\in\check{\nabla}_{G}\setminus\\{\vec{0},\vec{a}_{1},\ldots,\vec{a}_{n}\\}.$

Therefore, the exponent vectors associated with monomials with nonzero
coefficients in $\operatorname{init}_{\boldsymbol{v}}(\vec{h})$ are exactly
$\\{0,\vec{a}_{1},\ldots,\vec{a}_{n}\\}$. By Lemma 3.4,
$\vec{T}=\vec{G}_{\Delta}$ is an acyclic subdigraph and its associated
subgraph $T$ is a spanning tree of $G$. Indeed,
$\operatorname{init}_{\boldsymbol{v}}(\vec{h})$ consists of Laurent
polynomials

$\overline{w}_{i}-\sum_{j\in\mathcal{N}^{+}_{\vec{T}}(i)}a_{ij}\frac{x_{i}}{x_{j}}+\sum_{j\in\mathcal{N}^{-}_{\vec{T}}(i)}a_{ij}\frac{x_{j}}{x_{i}}\quad\text{for
}i=1,\ldots,n.$

By Lemma 3.5, this system has a unique $\mathbb{C}^{*}$-zero, which is
regular. Following from the principle of homotopy continuation, it gives rise
to a smooth curve defined by $\vec{h}(\boldsymbol{x},\tau e^{i\theta})=0$ in
$(\mathbb{C}^{*})^{n}\times(0,1]$.

This argument holds for every simplex in $\Delta_{\omega}$. That is, each
simplex $\Delta\in\Delta_{\omega}$ contributes a curve define by
$\vec{h}=\vec{0}$. Moreover, these curves are distinct, so the number of
curves $\eta$ satisfies
$\eta\geq|\Delta_{\omega}|=\operatorname{Vol}(\check{\nabla}_{G}),$ which
completes the proof.

This establishes the fact that the generic $\mathbb{C}^{*}$-zero count for the
algebraic Kuramoto system $\vec{f}_{G}$ is exactly the adjacency polytope
bound, despite the algebraic constraints on the parameters. Note that the BKK
bound for $\vec{f}_{G}$ is always between the generic $\mathbb{C}^{*}$-zero
count and the adjacency polytope bound. Therefore, we can indirectly derive
the Bernshtein-genericity of $\vec{f}_{G}$.

######  Corollary 3.9.

For generic real or complex $\overline{w}_{1},\ldots,\overline{w}_{n}$ and
generic but symmetric real or complex $k_{ij}=k_{ji}\neq 0$ for
$\\{i,j\\}\in\mathcal{E}(G)$, the algebraic Kuramoto system (3) is Bernshtein-
general, and

$\operatorname{MV}(\operatorname{Newt}(f_{G,1}),\ldots,\operatorname{Newt}(f_{G,n}))=\operatorname{Vol}(\operatorname{conv}(\operatorname{Newt}(f_{G,1})\cup\cdots\cup\operatorname{Newt}(f_{G,n})))=\operatorname{Vol}(\check{\nabla}_{G}).$

######  Remark 3.10.

We remark that the proof of Theorem 3.7 utilizes many ideas from the
polyhedral homotopy of Huber and Sturmfels [23]. The main differences are that
we consider a lifting function that preserves the relationships among the
coupling coefficients and we choose generators of the ideals at “toric
infinity” with a nice tree structure instead of requiring them to be binomial.

Finally, we draw attention to the tropical nature of this proof. The first
half of the proof establishes the tropical version of the generic root count
result: With the assignment of the valuation
$\operatorname{val}(\overline{w}_{i})=0$ and generic but symmetric
$\operatorname{val}(a_{ij})=\operatorname{val}(a_{ji})=\omega_{ij}=\omega_{ji}$,
we showed that the intersection number of the tropical hypersurfaces defined
by $f_{G,1},\ldots,f_{G,n}$ is bounded below by
$\operatorname{Vol}(\check{\nabla})$ which is also the stable self-
intersection number of the tropical hypersurface defined by the randomized
Kuramoto system (as defined in Section 2.6) with the same valuation. Computing
generic root count via stable tropical intersection is the topic of a recent
paper by Paul Helminck and Yue Ren [21], in which the special case for
complete networks is studied as an example.

Another consequence of Theorem 3.7 is the monotonicity of the generic
$\mathbb{C}^{*}$-zero count, since the volume of the adjacency polytope is
strictly increasing as new edges and nodes are added.

######  Corollary 3.11.

Let $(G_{1},K_{1},\vec{w}_{1})$ and $(G_{2},K_{2},\vec{w}_{2})$ be two
connected networks such that $G_{1}<G_{2}$. Then the generic
$\mathbb{C}^{*}$-zero count of $\vec{f}_{G_{1}}$ is strictly less than that of
$\vec{f}_{G_{2}}$.

In addition, we also get that the phase-delayed algebraic Kuramoto system and
lossy PV-type algebraic power flow system are Bernshtein-general.

######  Corollary 3.12.

For generic choices of $\overline{w}_{i},a_{ij},C_{ij}$, the algebraic
Kuramoto systems with phase delays (7) is Bernshtein-general, and its
$\mathbb{C}^{*}$-zero count equals to the normalized volume of $\nabla_{G}$.

######  Proof 3.13.

As noted before, for generic choices of
$\overline{w}_{1},\ldots,\overline{w}_{n}$, all $\mathbb{C}^{*}$-zeros of this
system are isolated and regular. Let $\eta$ be the generic number of
$\mathbb{C}^{*}$-zeros this system has. By Theorem 2.2 and Corollary 3.9, we
know $\eta\leq\operatorname{Vol}(\check{\nabla}_{G})$. By the Parameter
Homotopy Theorem [30], the number of $\mathbb{C}^{*}$-zeros of this
parametrized system constant on a nonempty Zariski open set of the parameters.
By setting $C_{ij}=1$, we recover (3) and by Theorem 3.7 and Corollary 3.9,
this system is Bernshtein-general, with
$\operatorname{Vol}(\check{\nabla}_{G})$ $\mathbb{C}^{*}$-zeros. This shows
that $\eta\geq\operatorname{Vol}(\check{\nabla}_{G})$, giving the result.

From the same argument, we get the generic zero count for the PV-type power
flow system.

######  Corollary 3.14.

For generic choices of $P_{i},b_{ij},g_{ij}$, the lossy PV-type algebraic
power flow system (6) is Bernshtein-general, and its $\mathbb{C}^{*}$-zero
count equals to the normalized volume of $\check{\nabla}_{G}$.

##  4 Explicit genericity conditions on coupling coefficients and refined zero
count

So far we have established that the generic $\mathbb{C}^{*}$-zero count of the
algebraic Kuramoto system $\vec{f}_{G}$ is the adjacency polytope bound
$\operatorname{Vol}(\check{\nabla}_{G})$. That is, for almost all choices of
the parameters $\vec{w}$ (natural frequencies) and $K$ (coupling
coefficients), the $\mathbb{C}^{*}$-zero count for $\vec{f}_{G}$ equals
$\operatorname{Vol}(\check{\nabla}_{G})$.

This section aims to understand when the $\mathbb{C}^{*}$-zero count drops
below the generic root count and by how much. We focus on the effect of
coupling coefficients $\\{k_{ij}\\}$ while leaving the choices of the natural
frequencies $\vec{w}$ generic, which ensures all $\mathbb{C}^{*}$-zeros of
$\vec{f}_{G}$ and $\vec{f}^{*}_{G}$ are isolated and regular.

######  Definition 4.1.

Given a nontrivial and connected graph $G$ with $n+1$ nodes, we define its
exceptional coupling coefficients $\mathcal{K}(G)$ to be the set of symmetric
and nonzero coupling coefficients $K=\\{k_{ij}\\}$ such that the number of
isolated $\mathbb{C}^{*}$-zeros of $\vec{f}_{(G,K,\vec{w})}=0$ is strictly
less than $\operatorname{Vol}(\check{\nabla}_{G})$, for any choice of
$\vec{w}\in\mathbb{C}^{n}$.

We first give an algebraic and graph-theoretic description for
$\mathcal{K}(G)$. Then, for certain families of graphs, we decompose
$\mathcal{K}(G)$ into strata, on which we develop refined root counts.

###  4.1 Exceptional coupling coefficients for a facial systems

We start with an analysis of a single facial system. For a given face $F$ of
$\check{\nabla}_{G}$, we describe the coupling coefficients for which the
facial system $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has a
$\mathbb{C}^{*}$-zero, signaling the genericity condition for Theorem 3.7 is
broken.

######  Definition 4.2.

Let $F\neq\varnothing$ be a proper face of $\check{\nabla}_{G}$ such that
$G_{F}$ is connected. We define the set

$\mathcal{K}(\vec{G}_{F})=\\{\boldsymbol{k}(\vec{G}_{F})\in(\mathbb{C}^{*})^{|F|}\mid\operatorname{init}_{F}(\vec{f}^{*}_{G})=\vec{0}\
\text{has a }\mathbb{C}^{*}\text{-solution}\\}$

to be the set of exceptional coupling coefficients with respect to the face
$F$ (or $\vec{G}_{F}$).

Recall that $\boldsymbol{k}(\vec{G}_{F})$ is the vector whose entries are the
coupling coefficients $k_{ij}$ for arcs $(i,j)$ in $\vec{G}_{F}$. By
Bernshtein’s Second Theorem (Theorem 2.2), if, for a proper face $F$ of
$\check{\nabla}_{G}$, the coupling coefficients
$\boldsymbol{k}(\vec{G}_{F})\in\mathcal{K}(\vec{G}_{F})$, then the algebraic
Kuramoto system $\vec{f}_{G}$ is not Bernshtein-general, and its isolated
$\mathbb{C}^{*}$-zero count is strictly less than the adjacency polytope bound
$\operatorname{Vol}(\check{\nabla}_{G})$. Note that since we fixed a single
face, this is inherently a local description of a patch of $\mathcal{K}(G)$
(Definition 4.1). By Theorems 3.7 and 2.2, $\mathcal{K}(\vec{G}_{F})$ is
contained in a proper and Zariski closed subset of Lebesgue measure
zero888This result is justified more directly in an unlabelled paragraph in
Bernshtein’s paper [3]. . In the following, we describe the structure of
$\mathcal{K}(\vec{G}_{F})$.

To lay down the algebraic foundation, we first show that every facial system
of $\vec{f}^{*}_{G}$ can be transformed into a “cycle form”, which can be
understood more easily using graph theoretic information.

######  Lemma 4.3.

Let $F\neq\varnothing$ be a proper face of $\check{\nabla}_{G}$,
$\check{Q}(\vec{G}_{F})$ and $\boldsymbol{a}(\vec{G}_{F})$ be the
corresponding reduced incidence matrix of $\vec{G}_{F}$ and its coupling
vector, respectively. Then
$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$ if and only
if

$\check{Q}(\vec{G}_{F})\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}=\vec{0}.$

This form will be referred to as the “cycle form” of the facial system
$\operatorname{init}_{F}(\vec{f}^{*}_{G})$ since the null space of the
incidence matrix of a digraph is spanned by fundamental cycle vectors.

######  Proof 4.4.

Recall that $\vec{f}^{*}_{G}=R\cdot\vec{f}_{G}$ for a nonsingular matrix
$R=[r_{ij}]$ (see Section 2.6). Since we assumed $a_{ij}=a_{ji}$, for each
$k=1,\dots,n$, we have

$\displaystyle\operatorname{init}_{F}(f^{*}_{G,k})(\boldsymbol{x})$
$\displaystyle=\sum_{(i,j)\in\vec{G}_{F}}(r_{ki}-r_{kj})\,a_{ij}\,\boldsymbol{x}^{\vec{e}_{i}-\vec{e}_{j}}=\sum_{(i,j)\in\vec{G}_{F}}\left\langle\,\boldsymbol{r}_{k}\,,\,\vec{e}_{i}-\vec{e}_{j}\,\right\rangle\,a_{ij}\,\boldsymbol{x}^{\vec{e}_{i}-\vec{e}_{j}},$

where $\boldsymbol{r}_{k}$ is the $k$-th row of the matrix $R$. Since vectors
of the form $\vec{e}_{i}-\vec{e}_{j}$ for $(i,j)\in\mathcal{E}(\vec{G}_{F})$
are exactly the columns in $\check{Q}(\vec{G}_{F})$,

$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=R\,\check{Q}(\vec{G}_{F})\,\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}.$

Since the square matrix $R$ is assumed to be nonsingular, this establishes the
equivalence.

This cycle form suggests a strong tie between the topological features of face
subgraphs of $G$ and the algebraic features of the initial systems of the
randomized Kuramoto system $\vec{f}^{*}_{G}$. The rest of this paper is
devoted to exploring this connection by leveraging the wealth of existing
knowledge about the facial complex of $\check{\nabla}_{G}$. In particular, we
will show that the cycle form of facial systems produces concrete genericity
conditions on the coupling coefficients.

Note that a facial system $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ can
always be decomposed into decoupled collections of facial systems
corresponding to weakly connected components of $\vec{G}_{F}$. We state this
as a lemma.

######  Lemma 4.5.

For a nonempty and proper face $F$ of $\check{\nabla}_{G}$ whose facial
subdigraph $\vec{G}_{F}$ consists of weakly connected components
$\vec{H}_{1},\ldots,\vec{H}_{\ell}$, there exists nonempty faces
$F_{1},\ldots,F_{\ell}$ of $F$ (which are also faces of $\check{\nabla}_{G}$)
such that $\vec{H}_{i}=\vec{G}_{F_{i}}$ for $i=1,\ldots,\ell$ and

$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}\quad\Longleftrightarrow\quad\operatorname{init}_{F_{i}}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}\quad\text{for
each }i=1,\ldots,\ell.$

######  Proof 4.6.

Let $V_{i}=\mathcal{V}(\vec{H}_{i})$ for $i=1,\ldots,\ell$ with $0\in V_{1}$.
We arrange the coordinates of $\mathbb{Z}^{n}$ according to the grouping
$V_{1},\ldots,V_{\ell}$ so that an inner normal vector
$\boldsymbol{\alpha}\in\mathbb{R}^{n}$ that defines the face $F$ of
$\check{\nabla}_{G}$ is written as
$\boldsymbol{\alpha}=\begin{bmatrix}\,\boldsymbol{\alpha}_{1}&\cdots&\boldsymbol{\alpha}_{\ell}\,\end{bmatrix}$
where $\boldsymbol{\alpha}_{i}\in\mathbb{R}^{|V_{i}|}$ corresponds to nodes in
$V_{i}$. Then for $i=1,\ldots,\ell$,

$\boldsymbol{v}_{i}:=\begin{bmatrix}\,\boldsymbol{0}_{|V_{1}|+\cdots+|V_{i-1}|}&\boldsymbol{\alpha}_{i}&\boldsymbol{0}_{|V_{i+1}|+\cdots+|V_{\ell}|}\,\end{bmatrix}\in\mathbb{R}^{n}$

defines a face $F_{i}$ of $\check{\nabla}_{G}$ such that
$\vec{H}_{i}=\vec{G}_{F_{i}}$.

By grouping the rows and columns of $\check{Q}(\vec{G}_{F})$ according to the
nodes and arcs in $H_{1},\ldots,H_{\ell}$, $\check{Q}(\vec{G}_{F})$ has the
block structure

$\check{Q}(\vec{G}_{F})=\left[\begin{smallmatrix}\check{Q}(\vec{H}_{1})&&\\\
&\ddots&\\\ &&\check{Q}(\vec{H}_{\ell})\end{smallmatrix}\right].$

Therefore, the cycle form of
$\operatorname{init}_{F}(\vec{f}^{*}_{G})=\vec{0}$, i.e.,
$\check{Q}(\vec{G}_{F})\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}=\vec{0},$
is equivalent to

$\check{Q}(\vec{H}_{i})\left(\boldsymbol{x}_{i}^{\check{Q}(\vec{H}_{i})}\circ\boldsymbol{a}(\vec{H}_{i})\right)^{\top}=\vec{0}\quad\text{for
each }i=1,\ldots,\ell,$

where $\boldsymbol{x}_{i}$ contain the coordinates corresponding to nodes in
$\vec{H}_{i}$. This is equivalent to
$\operatorname{init}_{F_{i}}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$ for
each $i=1,\ldots,\ell$, by Lemma 4.3.

Based on this observation, it is sufficient to consider facial systems
corresponding to connected facial subgraphs. We now give a description of the
coupling coefficients for which the facial system
$\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has a $\mathbb{C}^{*}$-zero.
Recall, a _bridge_ of a graph is an edge that is not contained in any cycle of
the graph. A graph that contains no bridges is said to be _bridgeless_. With
respect to a spanning tree $T$ of a connected graph, each edge outside of $T$
induces a unique cycle in this graph containing this edge. We call the
collection of such cycles the _fundamental cycles_ with respect to $T$.

######  Theorem 4.7.

Consider a corank-$d$ face $F$ of $\check{\nabla}_{G}$ such that $G_{F}$ is
connected.

  * •

If $G_{F}$ contains a bridge, then $\mathcal{K}(\vec{G}_{F})=\varnothing$.

  * •

If $G_{F}$ is bridgeless, for a choice of a spanning tree $T$ of $G_{F}$, let
$C_{1},\ldots,C_{d}<G_{F}$ be the fundamental cycles with respect to $T$ and
let $\vec{\eta}_{1},\ldots,\vec{\eta}_{d}$ be their associated primitive cycle
vectors with respect to $\vec{G}_{F}$. We define the very affine variety
$V(\vec{G}_{F})$ to be the set of
$(\boldsymbol{k},\boldsymbol{v})\in(\mathbb{C}^{*})^{2|F|}$ satisfying

$\left\\{\begin{aligned}
\boldsymbol{k}^{\vec{\eta}_{i}}&=\boldsymbol{v}^{\vec{\eta}_{i}}\quad\text{for
}i=1,\ldots,d.\\\
\boldsymbol{v}\,\check{Q}(\vec{G}_{F})^{\top}&=\boldsymbol{0}.\end{aligned}\right.$

Let $\pi:(\mathbb{C}^{*})^{2|F|}\to(\mathbb{C}^{*})^{|F|}$ be the projection
onto the first $|F|$ coordinates (i.e. the $\boldsymbol{k}$ coordinates). Then
$\mathcal{K}(\vec{G}_{F})=\pi(V(\vec{G}_{F}))$.

######  Proof 4.8.

First, we reduce the general statement to the special case in which $G_{F}$ is
spanning. After relabeling the nodes, we can assume
$\mathcal{V}(G_{F})=\\{0,\dots,n^{\prime}\\}$, and
$F\subset\mathbb{R}^{n^{\prime}}\times\\{\vec{0}\\}$. If $G_{F}$ is not
spanning, i.e., $n>n^{\prime}$, then

$\check{Q}(\vec{G}_{F})=\begin{bmatrix}\check{Q}(\vec{G}_{F^{\prime}})\\\
\boldsymbol{0}_{(n-n^{\prime})\times|F|}\end{bmatrix}.$

Thus the cycle form of the facial system
$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$, namely,

$\vec{0}=\check{Q}(\vec{G}_{F})\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}=\begin{bmatrix}\check{Q}(\vec{G}_{F^{\prime}})\\\
\boldsymbol{0}_{(n-n^{\prime})\times|F|}\end{bmatrix}\left(\boldsymbol{x}^{\left[\begin{smallmatrix}\check{Q}(\vec{G}_{F^{\prime}})\\\
\boldsymbol{0}_{(n-n^{\prime})\times|F|}\end{smallmatrix}\right]}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}$

is equivalent to

(9)
$\vec{0}=\check{Q}(\vec{G}_{F^{\prime}})\left((\boldsymbol{x}^{\prime})^{\check{Q}(\vec{G}_{F^{\prime}})}\circ\boldsymbol{a}(\vec{G}_{F^{\prime}})\right)^{\top}$

where $\boldsymbol{x}^{\prime}=(x_{1},\dots,x_{n^{\prime}})$ with
$(x_{n^{\prime}+1},\dots,x_{n})\in(\mathbb{C}^{*})^{n-n^{\prime}}$
unconstrained. This equation is, in turn, equivalent to the facet system
equation
$\operatorname{init}_{F^{\prime}}(\vec{f}^{*}_{G^{\prime}})(\boldsymbol{x}^{\prime})=\vec{0}$
by Lemma 4.3. Therefore, it is sufficient to assume that $n=n^{\prime}$, i.e.,
$G_{F}$ is spanning.

We now consider the first case: when $G_{F}$ contains a bridge. Since the null
space of $\check{Q}(\vec{G}_{F})$ is spanned by primitive cycle vectors, every
null vector must contain a zero coordinate, i.e.,
$\ker(\vec{G}_{F})\cap(\mathbb{C}^{*})^{|F|}=\varnothing$. Consequently,
$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$, which is
equivalent to

$\check{Q}(\vec{G}_{F})\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F^{\prime}})\right)^{\top}=\vec{0}$

has no $\mathbb{C}^{*}$-solutions for any choices
$\boldsymbol{a}(\vec{G}_{F})\in(\mathbb{C}^{*})^{|F|}$. This establishes the
first case.

For the second case, $G_{F}\leq G$ spans $G$. Therefore, a spanning tree $T$
of $G_{F}$ is also a spanning tree of $G$. Let $\vec{T}$ be the digraph
corresponding to $T$ such that $\vec{T}<=\vec{G}_{F}$, and by rearranging the
columns, we can write

$\displaystyle\check{Q}(\vec{G}_{F})$
$\displaystyle=\begin{bmatrix}\check{Q}(\vec{T})&\vec{v}_{1}&\cdots&\vec{v}_{d}\end{bmatrix}$
$\displaystyle\boldsymbol{a}(\vec{G}_{F})$
$\displaystyle=\begin{bmatrix}\boldsymbol{a}(\vec{T})&\alpha_{1}&\cdots&\alpha_{d}\end{bmatrix}$

where $\vec{v}_{1},\ldots,\vec{v}_{d}$ are the incidence vectors associated
with arcs of $\vec{G}_{F}$ that are outside $\vec{T}$, and
$\alpha_{1},\ldots,\alpha_{d}$ are their complex coupling coefficients. Let
$\vec{\eta}_{1},\ldots,\vec{\eta}_{d}\in\mathbb{Z}^{|F|}$ be the primitive
cycle vectors corresponding to the fundamental cycles induced by these arcs
with respect to $\vec{H}$. Since they are only determined up to a choice of
sign, we can assume

$\vec{\eta}_{i}=\begin{bmatrix}\vec{\tau}_{i}\\\
-\vec{e}_{i}\end{bmatrix}\quad\text{for some
}\vec{\tau}_{i}\in\\{-1,0,+1\\}^{n},$

in which $\vec{e}_{i}$ is considered as a vector in $\mathbb{Z}^{d}$. In other
words, the first $n$ coordinates (i.e., $\vec{\tau}_{i}$) of $\vec{\eta}_{i}$
are chosen so that $\vec{v}_{i}=\check{Q}(\vec{T})\,\tau_{i}$. By Lemma 2.4,
$\left\langle\,\boldsymbol{1}\,,\,\vec{\eta}_{i}\,\right\rangle=0$, therefore,
$\left\langle\,\boldsymbol{1}\,,\,\vec{\tau}_{i}\,\right\rangle=1$.

Suppose $\boldsymbol{k}(\vec{G}_{F})\in\pi(V(\vec{G}_{F}))$, as defined above,
we shall show $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has a
$\mathbb{C}^{*}$-zero. By assumption, there exists an
$\boldsymbol{\eta}=\begin{bmatrix}\boldsymbol{\tau}&u_{1}&\cdots&u_{d}\end{bmatrix}\in(\mathbb{C}^{*})^{n+d}$
such that $\boldsymbol{\eta}^{\top}\in\ker\check{Q}(\vec{G}_{F})$ and

$\boldsymbol{k}(\vec{G}_{F})^{\vec{\eta}_{i}}=(2\mathfrak{i}\cdot\boldsymbol{a}(\vec{G}_{F}))^{\vec{\eta}_{i}}=(2\mathfrak{i})^{\left\langle\,\boldsymbol{1}\,,\,\vec{\eta}_{i}\,\right\rangle}\circ\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta}_{i}}=\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta}_{i}}=\boldsymbol{\eta}^{\vec{\eta}_{i}}\quad\text{for
}i=1,\ldots,d.$

Note that with this partition of entries,
$\boldsymbol{\eta}^{\vec{\eta}_{i}}=\boldsymbol{\tau}^{\vec{\tau}_{i}}\cdot
u_{i}^{-1}$.

For any $\lambda\in\mathbb{C}^{*}$, define

$\boldsymbol{x}=(\lambda\cdot\boldsymbol{\tau}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}}\in(\mathbb{C}^{*})^{n}.$

Then from a straightforward calculation, we can verify that

$\displaystyle\boldsymbol{x}^{\vec{v}_{i}}$
$\displaystyle=\boldsymbol{x}^{\check{Q}(\vec{T})\vec{\tau}_{i}}=(\lambda\cdot\boldsymbol{\tau}\circ\boldsymbol{a}(\vec{T})^{-I})^{\vec{\tau}_{i}}.=\lambda\cdot\boldsymbol{\tau}^{\vec{\tau}_{i}}\circ\boldsymbol{a}(\vec{T})^{-\vec{\tau}_{i}}.=\lambda\,u_{i}\cdot\boldsymbol{\eta}^{\vec{\eta}_{i}}\circ\boldsymbol{a}(\vec{T})^{-\vec{\tau}_{i}},$

and thus

$\displaystyle\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})$
$\displaystyle=\begin{bmatrix}\boldsymbol{x}^{\check{Q}(\vec{T})}\circ\boldsymbol{a}(\vec{T})&\boldsymbol{x}^{\vec{v}_{1}}\cdot\alpha_{1}&\cdots&\boldsymbol{x}^{\vec{v}_{d}}\cdot\alpha_{d}\end{bmatrix}=\lambda\cdot\begin{bmatrix}\boldsymbol{\tau}&u_{1}&\cdots&u_{d}\end{bmatrix}=\lambda\,\boldsymbol{\eta}.$

That is,
$\check{Q}(\vec{G}_{F})(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F}))^{\top}=\vec{0}$.
By Lemma 4.3, the facial system $\operatorname{init}_{F}\vec{f}^{*}_{G}$ has a
$\mathbb{C}^{*}$-zero.

Conversely, suppose $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has a zero
$\boldsymbol{x}\in(\mathbb{C}^{*})^{n}$. By Lemma 4.3,

$\boldsymbol{\eta}:=(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F}))^{\top}\in\ker\check{Q}(\vec{G}_{F})\cap(\mathbb{C}^{*})^{|F|}.$

Since $\ker\check{Q}(\vec{G}_{F})$ is spanned by
$\vec{\eta}_{1},\ldots,\vec{\eta}_{d}$,
$\boldsymbol{\eta}=\sum_{j=1}^{d}\lambda_{j}\vec{\eta}_{j}^{\top}$ for some
$\lambda_{1},\ldots,\lambda_{d}\in\mathbb{C}$. This means

$\boldsymbol{\eta}^{\vec{\eta_{i}}}=(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F}))^{\vec{\eta_{i}}}=\boldsymbol{x}^{\check{Q}(\vec{G}_{F})\vec{\eta_{i}}}\circ\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta_{i}}}=\boldsymbol{x}^{\vec{0}}\circ\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta_{i}}}=\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta_{i}}}$

for $i=1,\ldots,d$, since $\vec{\eta}_{i}\in\ker\check{Q}(\vec{G}_{F})$. Also
recall that
$\boldsymbol{a}(\vec{G}_{F})=\frac{1}{2\mathfrak{i}}\boldsymbol{k}(\vec{G}_{F})$
and $\left\langle\,\mathbf{1}\,,\,\vec{\eta_{i}}\,\right\rangle=0$ for all
$i=1,\ldots,d$, so
$\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta}_{i}}=\boldsymbol{k}(\vec{G}_{F})^{\vec{\eta}_{i}}$
giving the result.

######  Remark 4.9.

This proof provides an explicit construction of an orbit in the
$\mathbb{C}^{*}$-zero set of the facial system
$\operatorname{init}_{F}(\vec{f}^{*}_{G})$ from which Section 5 will derive
interesting constructions.

######  Remark 4.10.

Since, in the context of Theorem 4.7, the null space of
$\check{Q}(\vec{G}_{F})$ is spanned by the primitive cycle vectors
$\vec{\eta}_{1},\ldots,\vec{\eta}_{d}$, the set $V(\vec{G}_{F})$ can also be
more conveniently be defined by

$\displaystyle\boldsymbol{k}^{\vec{\eta}_{i}}$
$\displaystyle=\left(\sum_{j=1}^{d}\lambda_{j}\vec{\eta}_{j}^{\top}\right)^{\vec{\eta}_{i}}\quad\text{for
}i=1,\ldots,d$ and
$\displaystyle\sum_{j=1}^{d}\lambda_{j}\vec{\eta}_{j}^{\top}$
$\displaystyle\in(\mathbb{C}^{*})^{|F|}$

with $\lambda_{1},\ldots,\lambda_{d}\in\mathbb{C}$ being part of the
coordinates. We will use the two versions interchangeably.

The complexity of the description of $\mathcal{K}(G_{F})$ amounts to the
complexity of computing the projection $\pi(V(\vec{G}_{F}))$. We now describe
cases where this projection can be computed explicitly.

A corank-0 face $F$ of $\check{\nabla}_{G}$ is a simplicial face, i.e., an
affinely independent face. Its corresponding facial subgraph $G_{F}$ must be a
forest and hence not bridgeless.

######  Corollary 4.11.

For a proper and nonempty corank-0 face $F$ of $\check{\nabla}_{G}$,
$\mathcal{K}(\vec{G}_{F})=\varnothing$, and the corresponding facial system
$\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has no $\mathbb{C}^{*}$-zero for
any choice of (nonzero) coupling coefficients.

We now consider the case of a facial subgraph consisting of $d$ cycles sharing
a single edge.

######  Proposition 4.12.

For an integer $d\geq 1$, let $F$ be a proper and nonempty corank-$d$ face of
$\check{\nabla}_{G}$ for which $G_{F}$ consists of of $d$ cycles
$\\{C_{1},\ldots,C_{d}\\}$ that share exactly one edge $e$ (i.e.,
$\mathcal{E}(C_{i})\cap\mathcal{E}(C_{j})=\\{e\\}$ for any pair of distinct
$i,j$), and let $\vec{\eta}_{1},\ldots,\vec{\eta}_{d}\in\\{-1,0,1\\}^{|F|}$ be
the primitive cycle vectors of $\vec{G}_{F}$ corresponding to
$C_{1},\ldots,C_{d}$ with consistent signs. Then
$\operatorname{init}_{F}(\vec{f}^{*})$ has a $\mathbb{C}^{*}$-zero if and only
if

(10)
$(-1)^{|C_{1}|/2}\boldsymbol{k}(\vec{G}_{F})^{\vec{\eta}_{1}}\,+\,\cdots\,+\,(-1)^{|C_{d}|/2}\boldsymbol{k}(\vec{G}_{F})^{\vec{\eta}_{d}}=1.$

Here, the “consistent” sign assignment means the common nonzero entries in
$\vec{\eta}_{1},\ldots,\vec{\eta}_{d}$ all share the same sign for $d>1$. If
$d=1$, the set $\\{\vec{\eta}_{1}\\}$ that contains the only choice of
primitive cycle vector, regardless of the signs, will still be considered as
consistent for simplicity.

######  Proof 4.13.

Recall that the face subgraph $G_{F}$ must be bipartite (Theorem 2.3) and
therefore the cycles $C_{1},\ldots,C_{d}$ must be even. For $i=1,\ldots,d$,
let $m_{i}=|C_{i}|$, $T_{i}=C_{i}-\\{e\\}$, and $\vec{T}_{i}<\vec{G}_{F}$ be
the corresponding subdigraph. We order the columns in the reduced incidence
matrix $\check{Q}(\vec{G}_{F})$ so that

$\check{Q}(\vec{G}_{F})=\begin{bmatrix}\check{Q}(\vec{T}_{1})&\cdots&\check{Q}(\vec{T}_{d})&\vec{v}_{e}\end{bmatrix},$

where $\vec{v}_{e}$ is the incidence vector associated with the edge $e$ in
$\vec{G}_{F}$. Since $e$ is contained $C_{1},\ldots,C_{d}$, there is a unique
vector $\vec{\tau}_{i}$, for each $i=1,\dots,d$, such that
$\vec{v}_{e}=\check{Q}(\vec{T}_{i})\,\vec{\tau}_{i}$. Since
$\operatorname{corank}(F)=d$, $\ker\check{Q}(\vec{G}_{F})$ is spanned by the
$d$ primitive cycle vectors (expressed as row vectors)

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34.44434pt\text{for }i=1,\ldots,d$

corresponding to the cycles $C_{1},\ldots,C_{d}$. Recall that, by Lemma 2.4,
$\left\langle\,\boldsymbol{1}\,,\,\vec{\eta}_{i}\,\right\rangle=0$, and thus
there must be $m_{i}/2$ positive entries and $m_{i}/2-1$ negative entries in
$\vec{\tau}_{i}$. That is,
$(\vec{\tau}_{i}^{\top})^{\vec{\tau}_{i}}=(-1)^{m_{i}/2-1}$.

Then any
$\boldsymbol{v}^{\top}\in\ker\check{Q}(\vec{G}_{F})\cap(\mathbb{C}^{*})^{|F|}$
can be expressed as

$\boldsymbol{v}=\sum_{i=1}^{d}\lambda_{i}\boldsymbol{\eta}_{i}=\begin{bmatrix}\lambda_{1}\vec{\tau}_{1}^{\top}&\cdots&\lambda_{d}\vec{\tau}_{d}^{\top}&-(\lambda_{1}+\cdots+\lambda_{d})\end{bmatrix}$

for some $\lambda_{i}\in\mathbb{C}^{*}$ such that
$\lambda_{1}+\cdots+\lambda_{d}\neq 0$. In this case, it is easy to verify
that

$\boldsymbol{v}^{\vec{\eta}_{i}}=\frac{(-1)^{m_{i}/2}\lambda_{i}}{\lambda_{1}+\cdots+\lambda_{d}}$

By Remark 4.10, the exceptional coupling coefficients
$\boldsymbol{k}:=\boldsymbol{k}(\vec{G}_{F})$ are defined by the equations

(11)
$(-1)^{m_{i}/2}\,\boldsymbol{k}^{\vec{\eta}_{i}}=\frac{\lambda_{i}}{\lambda_{1}+\ldots+\lambda_{d}}\quad\quad\text{for
}i=1,\ldots,d,$

and $\mathcal{K}(\vec{G}_{F})$ is the projection of the solution set of this
system onto the $\boldsymbol{k}$ coordinates. It suffices to show that this
projection is exactly the hypersurface, $\mathcal{H}$, defined by (10).

Summing equations (11) for $i=1,\ldots,d$, we get the equation (10) and
therefore, $\mathcal{K}(\vec{G}_{F})\subseteq\mathcal{H}$. Conversely, for any
$\boldsymbol{k}\in\mathcal{H}$, let
$\lambda_{i}=(-1)^{m_{i}/2}\boldsymbol{k}^{\vec{\eta}_{i}}\in\mathbb{C}^{*}$
for $i=1,\ldots,d$. Then $\lambda_{1}+\cdots+\lambda_{d}=1\neq 0$, and
$(-1)^{m_{i}/2}\boldsymbol{k}^{\vec{\eta}_{i}}=\frac{\lambda_{i}}{\lambda_{1}+\cdots+\lambda_{d}}$,
satisfying (11) so $\boldsymbol{k}\in\mathcal{K}(\vec{G}_{F})$.

In the special case of $d=1$ (i.e. $G_{F}$ contains a unique cycle) the above
proposition can be reduced to a particular simple binomial description for
$\mathcal{K}(\vec{G}_{F})$.

######  Corollary 4.14.

Let $F$ be a corank-1 face of $\check{\nabla}_{G}$ for which $G_{F}$ is
connected, $\vec{\eta}$ be a primitive cycle vector of $\vec{G}_{F}$, and
$\boldsymbol{k}(\vec{G}_{F})$ be the corresponding coupling vector. The facial
system $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ has a $\mathbb{C}^{*}$-zero
if and only if $G_{F}$ is a cycle and

(12) $\boldsymbol{k}(\vec{G}_{F})^{\vec{\eta}}=(-1)^{|F|/2}$

###  4.2 Global description of exceptional coupling coefficients

The set $\mathcal{K}(\vec{G}_{F})$ characterizes the coupling coefficients
that will cause a specific facial system of $\vec{f}^{*}_{G}$ to have
$\mathbb{C}^{*}$-zero. By Bernshtein’s Second Theorem (Theorem 2.2), the
$\mathbb{C}^{*}$-root count of $\vec{f}^{*}_{G}$ drops below the adjacency
polytope bound if and only if the coupling coefficients are in
$\mathcal{K}(\vec{G}_{F})$ for some nonempty and proper face $F$ of
$\check{\nabla}_{G}$. By taking the union of all such faces, this gives a
global description of the exceptional coupling coefficients $\mathcal{K}(G)$
for which the genericity condition in Theorem 3.7 is broken.

Note, however, that most faces have no contribution to the set of exceptional
coupling coefficients. Combining Theorem 2.3 and Theorem 4.7, we get the
following topological constraints on the faces with potentially nontrivial
contribution to $\mathcal{K}(G)$.

######  Proposition 4.15.

Let $\Phi$ be the subset of nontrivial and proper faces of
$\check{\nabla}_{G}$ such that their facial subgraphs are

  1. 1.

bridgeless (and hence cyclic), and

  2. 2.

maximally bipartite in their induced subgraphs.

Then the set of exceptional coupling coefficients is the union

$\mathcal{K}(G)=\bigcup_{F\in\Phi}\mathcal{K}(\vec{G}_{F}).$

In this union, subsets $\mathcal{K}(\vec{G}_{F})$ are assumed to be embedded
in the common ambient space that contains all coupling coefficients.

Based on this and the local description from Proposition 4.12, we can derive
the global description of the set of exceptional coupling coefficients for
graphs consisting of multiple cycles sharing a single edge. The geometry of
the adjacency polytope for such networks was studied in detail by D’Ali,
Delucchi, and Michałek [17] and the closely related work [14].

As noted in the proof of Proposition 4.12, it is sufficient to focus on even
cycles, since odd cycles have no contributions to the exceptional coupling
coefficients.

######  Proposition 4.16.

Let $G$ be a graph consisting of $d$ independent even cycles,
$C_{1},\ldots,C_{d}$ that overlap at a single edge $e$ (i.e.,
$\mathcal{E}(C_{i})\cap\mathcal{E}(C_{j})=\\{e\\}$ for any $i\neq j$). Let
$U_{i}\subset\\{-1,0,+1\\}^{|\mathcal{E}(G)|}$, for $i=1,\ldots,d$ be the set
of balanced primitive cycle vectors associated with $C_{1},\ldots,C_{d}$. Then
the set of exceptional coupling coefficients $\mathcal{K}(G)$ is the union of
subsets

(13)
$\left\\{\boldsymbol{k}\;:\;\sum_{j=1}^{d}(-1)^{|C_{i_{j}}|/2}\boldsymbol{k}^{\vec{\eta}_{i_{j}}}=1\text{
where }(\vec{\eta}_{i_{1}},\ldots,\vec{\eta}_{i_{t}})\in
U_{i_{1}}\times\cdots\times U_{i_{t}}\text{ is consistent}\right\\}.$

Here, _balanced_ means each $\vec{\eta}_{i_{j}}$ satisfies
$\left\langle\,\boldsymbol{1}\,,\,\vec{\eta}_{i_{j}}\,\right\rangle=0$ (has
equal numbers of $-1$ and $+1$). And the tuple
$(\vec{\eta}_{i_{1}},\ldots,\vec{\eta}_{i_{t}})$ is considered _consistent_ if
$\\{i_{1},\ldots,i_{t}\\}$ are distinct and for each pair $i_{j}$ and
$i_{j^{\prime}}$, the common nonzero entries of $\vec{\eta}_{i_{j}}$ and
$\vec{\eta}_{i_{j^{\prime}}}$ have the same signs, as in Proposition 4.12.

######  Proof 4.17.

By Proposition 4.15, the only contributions to $\mathcal{K}(G)$ come from
nonempty proper faces associated with face subgraphs that are bridgeless and
maximally bipartite in their induced subgraphs. Let $\Phi$ be the set of such
faces (as in Proposition 4.15). In this case, these are face subgraphs
consisting of exactly a subset of the cycles $C_{1},\ldots,C_{d}$.

It suffices to show that consistent tuples
$(\vec{\eta}_{i_{1}},\ldots,\vec{\eta}_{i_{t}})\in U_{i_{1}}\times\cdots\times
U_{i_{t}}$ produce the primitive cycle vectors for all facial subdigraphs
$\\{\vec{G}_{F}\mid F\in\Phi\\}$, since, by Proposition 4.12, each subset is
simply $\mathcal{K}(\vec{G}_{F})$ in that case. This is given by [12, Theorem
7] which says that for each $(\vec{\eta}_{i_{1}},\ldots,\vec{\eta}_{i_{t}})\in
U_{i_{1}}\times\cdots\times U_{i_{t}}$ there exists a corank-$t$ face $F$ of
$\check{\nabla}_{G}$ such that $\vec{\eta}_{i_{1}},\ldots,\vec{\eta}_{i_{t}}$
are the primitive cycle vectors of $\vec{G}_{F}$, and vice versa. With this,
the statement is an implication of Proposition 4.12 and Proposition 4.15.

###  4.3 Refined root count for unicycle networks

$G$ is a unicycle graph if it contains a unique cycle. For such a graph,
Corollary 4.14 can be sharpened into a condition that detects and quantifies
the drop in the $\mathbb{C}^{*}$-root count of $\vec{f}_{G}$ relative to the
adjacency polytope bound.

Intuitively speaking, all coupling coefficients in $\mathcal{K}(G)$ are
exceptional, in the sense that they cause the $\mathbb{C}^{*}$-zero count of
the algebraic Kuramoto system to drop below the adjacency polytope bound.
However, some are “more exceptional” than others. In this section, we develop
this idea rigorously. We begin by restating the definition of the exceptional
coupling coefficients with respect to a face in terms of network properties
under the unicycle assumption.

######  Definition 4.18 (Balanced subnetwork).

For a unicycle graph $G$ that contains a unique cycle $O$, let $\vec{O}^{+}$
and $\vec{O}^{-}$ be digraphs corresponding to assigning clockwise and
counterclockwise orientations to $O$, respectively. For an acyclic subdigraph
$\vec{H}<\vec{G}$, we say $(\vec{H},K)$ is a balanced directed acyclic
subnetwork of $(G,K)$ if

  * •

$|\mathcal{E}(\vec{H})\cap\mathcal{E}(\vec{O}^{+})|=|\mathcal{E}(\vec{H})\cap\mathcal{E}(\vec{O}^{-})|=|\mathcal{V}(O)|/2$;
and

  * •

let $\kappa^{+}(\vec{H})$ and $\kappa^{-}(\vec{H})$ be the products of the
coupling coefficients along the arcs in
$\mathcal{E}(\vec{H})\cap\mathcal{E}(\vec{O}^{+})$ and
$\mathcal{E}(\vec{H})\cap\mathcal{E}(\vec{O}^{-})$, respectively, then

$\frac{\kappa^{+}(\vec{H})}{\kappa^{-}(\vec{H})}=(-1)^{|\mathcal{V}(O)|/2}.$

$0$$2$$1$$3$$k_{10}$$k_{12}$$k_{30}$$k_{32}$ Figure 1: A balanced subnetwork
of a 4-cycle

Since the only type of subnetworks of interest here are directed and acyclic,
they will simply be referred to as “balanced subnetworks”. Note that this
definition implies that the unique cycle $O$ must be an even cycle. For
example, the subdigraph $\vec{H}$ of $C_{4}$ shown in Figure 1 is acyclic and
contains two clockwise arcs $k_{10},k_{32}$ and two counterclockwise arcs
$k_{12},k_{30}$. Thus, $\kappa^{+}(\vec{H})=k_{10}k_{32}$ and
$\kappa^{-}(\vec{H})=k_{12}k_{30}$, and it is considered a balanced subnetwork
if $k_{10}k_{32}=k_{12}k_{30}$. Using this concept Corollary 4.14 can be
restated into the following.

######  Proposition 4.19.

Let $(G,K,\vec{w})$ be a network in which $G$ is a connected unicycle graph.
For generic real or complex $\vec{w}$, the following are equivalent:

  1. 1.

the $\mathbb{C}^{*}$-zero count of $\vec{f}_{G}$ is strictly less than
$\operatorname{Vol}(\check{\nabla}_{G})$;

  2. 2.

$(G,K,\vec{w})$ has a balanced directed acyclic subnetwork.

If the unique cycle in $G$ is an odd cycle, then there can be no balanced
subnetworks, and thus the $\mathbb{C}^{*}$-zero count of $\vec{f}_{G}$ must
reach the adjacency polytope bound $\operatorname{Vol}(\check{\nabla}_{G})$.
Combined with the known results on adjacency polytope bound [13], we obtain
the following stronger root count.

######  Corollary 4.20.

Let $(G,K,\vec{w})$ be a network in which $G$ is a connected unicycle graph
and its unique cycle $O$ is an odd cycle. Then for any choices of real or
complex $\vec{w}$ and
$K=\\{k_{ij}\in\mathbb{C}^{*}\mid\\{i,j\\}\in\mathcal{E}(G)\\}$, the
$\mathbb{C}^{*}$ zeroes of $\vec{f}_{G}$ are all isolated, and the total
number, counting multiplicity, equals the adjacency polytope bound

$\operatorname{Vol}(\check{\nabla}_{G})=2^{\ell}\,m\,\binom{m-1}{(m-1)/2},$

where $m=|\mathcal{V}(O)|$ and $\ell=|\mathcal{V}(G)\setminus\mathcal{V}(O)|$.

If the unique cycle $O$ in $G$ is even, then there may be balanced
subnetworks. The usefulness of the concept of balanced subnetworks lies in the
fact that each maximal balanced subnetwork reduces the $\mathbb{C}^{*}$-zero
count of $\vec{f}_{G}$ by one. Here, maximal balanced subnetworks are simply
maximal elements in the poset of balanced subnetworks. To establish this, we
will utilize the facet decomposition homotopy [9, Theorem 3], which we
summarize as the following lemma with minor modifications.

######  Lemma 4.21 (Adapted from Theorem 3 [9]).

For generic choices of real or complex $\vec{w}$ and symmetric, nonzero
coupling coefficients $K=\\{k_{ij}\mid\\{i,j\\}\in\mathcal{E}(G)\\}$, let
$\vec{f}_{G}$ be the algebraic Kuramoto system derived from $(G,K,\vec{w})$.
Then the number of isolated $\mathbb{C}^{*}$-zeros that $\vec{f}_{G}$ has
equals to the sum over all facets $F\in\mathcal{F}(\check{\nabla}_{G})$ of the
$\mathbb{C}^{*}$-zero count for pyramid systems of the form

(14)
$\check{Q}(\vec{G}_{F})\left(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F})\right)^{\top}=\vec{w}.$

We call the above system a _pyramid system_ 999In [9], the first named author
named such a system a “facet system”. This name, unfortunately, conflicts with
the terms facial system and facet system we adopt here.  because the Newton
polytope of this system is a pyramid of the form
$\operatorname{conv}(\\{\vec{0}\\}\cup F)$.

######  Proposition 4.22.

Let $(G,K,\vec{w})$ be a network in which $G$ is a connected unicycle graph,
and its unique $m$-cycle $O$ is an even cycle. Then for generic choices of
real or complex $\vec{w}$, the $\mathbb{C}^{*}$-zeros of $\vec{f}_{G}$ are all
isolated, and the total number equals

$2^{\ell}\,\left[m\binom{m-1}{m/2-1}-\beta\right],$

where $\ell=|\mathcal{V}(G)\setminus\mathcal{V}(O)|$, and $\beta$ is the
number of maximally balanced subnetwork in $(G,K,\vec{w})$.

######  Proof 4.23.

By Lemma 2.1 and an induction on leaf nodes, it is sufficient to assume
$G=C_{N}$ and show the $\mathbb{C}^{*}$-zero count to be
$N\binom{N-1}{N/2-1}-\beta$. Indeed, it was shown in [13, Theorem 13] that the
adjacency polytope bound, i.e., the generic $\mathbb{C}^{*}$-zero count is
$N\binom{N-1}{N/2-1}$. So we simply have to show each maximally balanced
subnetwork reduces the $\mathbb{C}^{*}$-zero count by 1.

Fix a facet $F\in\mathcal{F}(\check{\nabla}_{C_{N}})$ that corresponds to
$\vec{H}=(\vec{C}_{N})_{F}$ in a maximally balanced subnetwork
$(\vec{H},K,\vec{w})$. That is, we assume

$(-1)^{N/2}=\frac{\kappa^{+}(\vec{H})}{\kappa^{-}(\vec{H})}=\boldsymbol{a}(\vec{H})^{\vec{\eta}},$

where $\vec{\eta}$ is a primitive cycle vector of $\vec{H}$. We shall first
show that under this assumption the number of $\mathbb{C}^{*}$-solutions to
the pyramid system (14) has is exactly $N/2-1$ (which is one less than the
generic $\mathbb{C}^{*}$-zero count of the pyramid system $N/2$ as shown in
[13, Proposition 12].)

We start by rewriting (14) in a convenient form. Fix an arbitrary arc $e$ of
$\vec{H}$, and let $\vec{T}=\vec{H}-\\{e\\}$, then the corresponding graph $T$
is a spanning tree of $C_{N}$, and therefore $\check{Q}(\vec{T})$ is
nonsingular. We shall arrange the entries so that
$\check{Q}(\vec{H})=\begin{bmatrix}\,\check{Q}(\vec{T})&\vec{v}_{e}\,\end{bmatrix}$
and
$\boldsymbol{a}(\vec{H})=\begin{bmatrix}\,\boldsymbol{a}(\vec{T})&a_{e}\,\end{bmatrix}$,
and where $\vec{v}_{e}$ and $a_{e}$ are the incidence vector and the complex
coupling coefficients for the arc $e$, respectively. By choosing an
orientation for the cycle, we can assume the last coordinate of the primitive
cycle vector $\boldsymbol{\eta}$ is $-1$. That is,
$\boldsymbol{\eta}=\begin{bmatrix}\boldsymbol{\eta}_{\vec{T}}&-1\end{bmatrix}$,
where $\boldsymbol{\eta}_{\vec{T}}$ is the incidence vector of the oriented
cycle with respect to the arcs in $\vec{T}$. Let
$\boldsymbol{y}=(\check{Q}(\vec{T})^{-1}\vec{w})^{\top}$, and $y_{n+1}=0$,
which can be assumed to be generic since $\vec{w}$ is generic. Then the
pyramid system (14) is equivalent to

(15)
$\displaystyle\boldsymbol{x}^{\check{Q}(\vec{T})}\circ\boldsymbol{a}(\vec{T})$
$\displaystyle=\boldsymbol{y}+\lambda\boldsymbol{\eta}_{\vec{T}}$ (16)
$\displaystyle\boldsymbol{x}^{\vec{v}_{e}}\cdot a_{e}$
$\displaystyle=y_{n+1}-\lambda.$

Recall that $\boldsymbol{a}(\vec{T})\in(\mathbb{C}^{*})^{n}$, so (15) is
equivalent to

$\boldsymbol{x}^{\check{Q}(\vec{T})}=\boldsymbol{a}(\vec{T})^{-I}\circ(\boldsymbol{y}+\lambda\boldsymbol{\eta}_{\vec{T}}).$

Moreover, since $\check{Q}(\vec{T})$ is unimodular, it has a unique
$\mathbb{C}^{*}$-solution for any given $\lambda\in\mathbb{C}$, as long as the
right-hand-side is also in $(\mathbb{C}^{*})^{n}$. That is, under this
assumption, the value of $\boldsymbol{x}$, in a solution, is uniquely
determined by the value of $\lambda$. Therefore, it is sufficient to solve for
$\lambda$ and count the distinct solutions.

Note that $\vec{v}_{e}=\check{Q}(\vec{T})\,\vec{\eta}_{\vec{T}}$, so
substituting (15) into (16) produces

$y_{n+1}-\lambda=\boldsymbol{x}^{\vec{v}_{e}}\cdot
a_{e}=(\boldsymbol{x}^{\check{Q}(\vec{T})})^{\vec{\eta}_{\vec{T}}}\cdot
a_{e}=(\boldsymbol{a}(\vec{T})^{-I}\circ(\boldsymbol{y}+\lambda\boldsymbol{\eta}_{\vec{T}}))^{\vec{\eta}_{\vec{T}}}\cdot
a_{e}=\boldsymbol{a}(\vec{G}_{F})^{-\vec{\eta}}\cdot(\boldsymbol{y}+\lambda\boldsymbol{\eta}_{\vec{T}})^{\vec{\eta}_{\vec{T}}}$

The problem is now reduced to the problem of counting the number of
$\lambda\in\mathbb{C}$ that satisfies this equation. Let
$I^{+}=\\{i\in[n]\mid\eta_{i}=+1\\}$ and $I^{-}=\\{i\in[n]\mid\eta_{i}=-1\\}$,
respectively. Then, the above equation is equivalent to

$\prod_{i\in
I^{-}}(y_{i}-\lambda)=\boldsymbol{a}(\vec{G}_{F})^{-\vec{\eta}}\prod_{i\in
I^{+}}(y_{i}+\lambda),$

which is a polynomial equation in $\lambda$ of degree up to $N/2$, i.e., up to
$N/2$ solutions for $\lambda$. However, we will show this upper bound is not
attainable.

Note that the degree $N/2$ terms of the two sides of the above equation are
$(-1)^{N/2}\lambda^{N/2}$ and
$\boldsymbol{a}(\vec{G}_{F})^{-\vec{\eta}}\lambda^{N/2}$, respectively. Since
we assumed that $\boldsymbol{a}(\vec{G}_{F})^{\vec{\eta}}=(-1)^{N/2}$, the
degree $N/2$ terms cancel, and therefore the above equation is actually a
polynomial equation in $\lambda$ of degree up to $N/2-1$ with the coefficients
being symmetric functions in $\boldsymbol{y}$, which are assumed to be
generic. Consequently, there are exactly $N/2-1$ solutions to this equation in
$\lambda$, which produces $N/2-1$ distinct solutions to the system (15)-(16).
That is, the pyramid system induced by the facet $F$ has $N/2-1$ distinct
$\mathbb{C}^{*}$-zeros.

So far, we have shown that, for any facet associated with a maximally balanced
subnetwork, the pyramid system has $N/2-1$ $\mathbb{C}^{*}$-solutions, whereas
a pyramid system induced by other facets has $N/2$ $\mathbb{C}^{*}$-solutions.
It was established in [13, Proof of Theorem 13] that $\check{\nabla}_{C_{N}}$
has $2\binom{N-1}{N/2-1}$ facets. Therefore, the sum of the number of
$\mathbb{C}^{*}$-solutions to each pyramid system defined in (14) is:

$\beta\,\left(\frac{N}{2}-1\right)+\left(2\binom{N-1}{N/2-1}-\beta\right)\,\frac{N}{2}=N\binom{N-1}{N/2-1}-\beta.$

An application of Lemma 4.21 then gives the result.

$0$$2$$1$$3$$k_{10}$$k_{12}$$k_{30}$$k_{32}$

$0$$2$$1$$3$$k_{10}$$k_{12}$$k_{30}$$k_{32}$

$0$$2$$1$$3$$k_{10}$$k_{12}$$k_{30}$$k_{32}$

Figure 2: Representatives of the classes of balanced subnetworks in $C_{4}$

######  Example 4.24 (The 4-cycle case).

Consider the case of $G=C_{4}$. As shown in [13, Theorem 16] and Theorem 3.7,
for generic choices of $\vec{w}$ and $K$, $\vec{f}_{G}$ has 12
$\mathbb{C}^{*}$-zeros. This is the adjacency polytope bound. Now, with the
choice of $\vec{w}$ remain generic, Proposition 4.22 provides a stratification
of the space of coupling coefficients according to the maximum
$\mathbb{C}^{*}$-zero count.

This network has up 6 balanced subnetworks supported by the digraphs shown in
Figure 2 and their transposes. With these, we have the following decomposition
of the coupling coefficients:

  1. 1.

Consider the 1-dimensional “balancing variety” defined by the binomial system

$\left\\{\begin{aligned} k_{10}k_{12}k_{32}^{-1}k_{30}^{-1}&=1\\\
k_{10}k_{12}^{-1}k_{32}^{-1}k_{30}&=1\\\
k_{10}k_{12}^{-1}k_{32}k_{30}^{-1}&=1,\end{aligned}\right.$

which will produce all 6 balanced subnetworks. It can be parametrized by

$k_{10}=k_{12}=k_{32}=k_{30}=s,$

for $s\in\mathbb{C}^{*}$. Any choice of $s\in\mathbb{C}^{*}$ produces an
algebraic Kuramoto system $\vec{f}_{G}$ with at most $12-6=6$
$\mathbb{C}^{*}$-zeros. For generic $s$, the $\mathbb{C}^{*}$-zero count is
exactly 6. Moreover, taking $s=1$ and
$\omega_{1}=1.1,\omega_{2}=-2.1,\omega_{3}=1$, all six $\mathbb{C}^{*}$-zeros
can be _real_ (see Appendix B).

  2. 2.

There are three 2-dimensional balancing varieties defined by two out of three
binomial equations from the above system:

$\displaystyle\left\\{\begin{aligned} k_{10}k_{12}k_{32}^{-1}k_{30}^{-1}&=1\\\
k_{10}k_{12}^{-1}k_{32}^{-1}k_{30}&=1\end{aligned}\right.$
$\displaystyle\left\\{\begin{aligned} k_{10}k_{12}^{-1}k_{32}^{-1}k_{30}&=1\\\
k_{10}k_{12}^{-1}k_{32}k_{30}^{-1}&=1,\end{aligned}\right.$
$\displaystyle\left\\{\begin{aligned} k_{10}k_{12}k_{32}^{-1}k_{30}^{-1}&=1\\\
k_{10}k_{12}^{-1}k_{32}k_{30}^{-1}&=1,\end{aligned}\right.$

each producing 4 balanced subnetworks. They contain choices of coupling
coefficients for which $\vec{f}_{G}$ has at most 8 $\mathbb{C}^{*}$-zeros. A
generic choice of coupling coefficients on this variety will produce exactly 8
$\mathbb{C}^{*}$-zeros. The first of the three, can be parametrized by

$\displaystyle k_{10}$ $\displaystyle=s$ $\displaystyle k_{12}$
$\displaystyle=t$ $\displaystyle k_{32}$ $\displaystyle=\pm t$ $\displaystyle
k_{30}$ $\displaystyle=\pm s,$

where $(s,t)\in(\mathbb{C}^{*})^{2}$. The other two,
$\mathcal{K}(\vec{\eta}_{2},\vec{\eta}_{3})$ and
$\mathcal{K}(\vec{\eta}_{3},\vec{\eta}_{1})$, can be parametrized similarly.
Moreover, taking $s=1,t=-1.001,\omega_{1}=1.1,\omega_{2}=-2.1,\omega_{3}=1$
all $8$ solutions can be real.

  3. 3.

There are three 3-dimensional balancing varieties defined by binomial
equations

$\displaystyle k_{10}k_{12}k_{32}^{-1}k_{30}^{-1}$ $\displaystyle=1$
$\displaystyle k_{10}k_{12}^{-1}k_{32}^{-1}k_{30}$ $\displaystyle=1$
$\displaystyle k_{10}k_{12}^{-1}k_{32}k_{30}^{-1}$ $\displaystyle=1,$

each producing 2 balanced subnetworks. They contain coupling coefficients for
which $\vec{f}_{G}$ has at most 10 $\mathbb{C}^{*}$-zeros. A generic choice of
coupling coefficients on one of these varieties gives exactly 10
$\mathbb{C}^{*}$-zeros. The first of the three can be parametrized by

$\displaystyle k_{10}$ $\displaystyle=s$ $\displaystyle k_{12}$
$\displaystyle=u/s$ $\displaystyle k_{32}$ $\displaystyle=t$ $\displaystyle
k_{30}$ $\displaystyle=u/t,$

for $(s,t,u)\in(\mathbb{C}^{*})^{3}$. The other two can be parametrized
similarly. 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.

  4. 4.

The remaining choices of $K$, form a Zariski-dense subset in the space of all
coupling coefficients for $C_{4}$, and define an algebraic Kuramoto system
with 12 $\mathbb{C}^{*}$-zeros.

Under the assumption that $\vec{w}$ is generic, there are no other
possibilities.

######  Example 4.25 (The 6-cycle case).

Similarly, for the case of $G=O=C_{6}$, it is known that for generic choices
of $\vec{w}$ and $K$, $\vec{f}_{G}$ has 60 $\mathbb{C}^{*}$-zeros. There are
up to 20 balanced subnetworks that form 10 transpose pairs, and they can be
described by arc orientation vectors $\vec{\eta}_{1},\ldots,\vec{\eta}_{10}$:

$\displaystyle\left[\begin{smallmatrix}+1\\\ +1\\\ +1\\\ -1\\\ -1\\\
-1\end{smallmatrix}\right],$ $\displaystyle\left[\begin{smallmatrix}+1\\\
+1\\\ -1\\\ +1\\\ -1\\\ -1\end{smallmatrix}\right],$
$\displaystyle\left[\begin{smallmatrix}+1\\\ +1\\\ -1\\\ -1\\\ +1\\\
-1\end{smallmatrix}\right],$ $\displaystyle\left[\begin{smallmatrix}+1\\\
+1\\\ -1\\\ -1\\\ -1\\\ +1\end{smallmatrix}\right],$
$\displaystyle\left[\begin{smallmatrix}+1\\\ -1\\\ +1\\\ +1\\\ -1\\\
-1\end{smallmatrix}\right],$ $\displaystyle\left[\begin{smallmatrix}+1\\\
-1\\\ +1\\\ -1\\\ +1\\\ -1\end{smallmatrix}\right],$
$\displaystyle\left[\begin{smallmatrix}+1\\\ -1\\\ +1\\\ -1\\\ -1\\\
+1\end{smallmatrix}\right],$ $\displaystyle\left[\begin{smallmatrix}+1\\\
-1\\\ -1\\\ +1\\\ +1\\\ -1\end{smallmatrix}\right],$
$\displaystyle\left[\begin{smallmatrix}+1\\\ -1\\\ -1\\\ +1\\\ -1\\\
+1\end{smallmatrix}\right],$ $\displaystyle\left[\begin{smallmatrix}+1\\\
-1\\\ -1\\\ -1\\\ +1\\\ +1\end{smallmatrix}\right],$

where the ordering of the coordinates corresponds to the ordering
$\boldsymbol{k}(O)=[k_{01},k_{12},k_{23},k_{34},k_{45},k_{50}]$. The
exceptional coupling coefficients for which the $\mathbb{C}^{*}$-root count of
$\vec{f}_{G}$ deviates from the generic root count are contained in the zero
sets of binomial equations of the form

$\boldsymbol{k}(O)^{\vec{\eta}_{i}}=-1,$

for $i\in[10]$. This indicates that the exceptional coupling coefficients must
include some negative values, which corresponds to repulsive couplings. The
full description of the strata of exceptional coupling coefficients will be
lengthy, so we only examine two special case.

Consider the family of coupling coefficients

$\boldsymbol{k}(O)=[\pm s,\ldots,\pm s],$

with an odd number of negative choices, parametrized by $s\in\mathbb{C}^{*}$.
Then $\boldsymbol{k}(O)^{\vec{\eta}_{i}}=-1$ is satisfied for $i=1,\ldots,10$.
By Proposition 4.22, for any $s\in\mathbb{C}^{*}$ and generic choices of
$\vec{w}$, the $\mathbb{C}^{*}$-zero count of $\vec{f}_{G}$ is no more than
$6\cdot\binom{6-1}{6/2-1}-2\cdot 10=40$. This bound is exact for generic
choice of $s\in\mathbb{C}^{*}$.

Similarly, for the family

$\boldsymbol{k}(O)=[\pm s,\pm s,\pm t,\ldots,\pm t]$

with an odd number of negative choices, parametrized by
$(s,t)\in(\mathbb{C}^{*})^{2}$, the conditions
$\boldsymbol{k}(O)^{\vec{\eta}_{i}}=-1$ is satisfied only for
$i=5,6,7,8,9,10$. Therefore, the $\mathbb{C}^{*}$-zero count of $\vec{f}_{G}$
is no more than $6\cdot\binom{6-1}{6/2-1}-2\cdot 6=48$ for any
$(s,t)\in(\mathbb{C}^{*})^{2}$ and generic $\vec{w}$. It is exact for generic
choices of $(s,t)$.

##  5 Positive-dimensional synchronization configurations for bipartite
networks

We now turn our attention to positive-dimensional zero sets of the algebraic
Kuramoto equations. They represent synchronization configurations that have at
least one degree of freedom.

We generalize existing constructions of positive-dimensional zero sets for the
Kuramoto equations [1, 16, 28, 32] and characterize conditions under which
they arise.

The crucial observation that enables our constructions is that when $G$ is
bipartite, by Theorem 2.3, $G$ itself is a facet subgraph, i.e., there exists
a $F\in\mathcal{F}(\check{\nabla}_{G})$ such that $G=G_{F}$. Indeed,

$\check{\nabla}_{G}=\\{0\\}\cup F\cup(-F).$

That is, $F$ and $-F$ contains the exponent vectors of all nonconstant terms.
If we further assume the oscillators are homogeneous, i.e., all the natural
frequencies are identical and thus the constant terms in $\vec{f}_{G}$ are all
zero, then the algebraic Kuramoto system can be expressed in the simple form

$\vec{f}_{G}(\boldsymbol{x})=\check{Q}(G)(\boldsymbol{x}^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(G))^{\top}-\check{Q}(G)(\boldsymbol{x}^{-\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(G))^{\top}.$

Note that both terms are cycle forms of facial systems, as shown in Lemma 4.3,
and they correspond to $\operatorname{init}_{F}(\vec{f}^{*}_{G})$ and
$\operatorname{init}_{-F}(\vec{f}^{*}_{G})$, respectively. In the previous
section, we already constructed explicit solutions to
$\operatorname{init}_{F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$ under
certain restrictions of the coupling coefficients. In the following, we
investigate additional restrictions that will ensure the same solution also
satisfies $\operatorname{init}_{-F}(\vec{f}^{*}_{G})(\boldsymbol{x})=\vec{0}$
and will therefore be a positive dimensional $\mathbb{C}^{*}$-zero set to
$\vec{f}_{G}$ itself.

###  5.1 Unicycle graphs

We first consider the case when $G$ is a unicycle graph. In this context,
Proposition 4.19 can be paraphrased as the following necessary condition for
the existence positive-dimensional $\mathbb{C}^{*}$-zero sets.

######  Proposition 5.1.

Let $G$ be a (connected) unicycle graph that contains a unique cycle $O$. If
the $\mathbb{C}^{*}$-zero sets of $\vec{f}_{G}$ is positive-dimensional, then
$O$ is even and $(G,K)$ contains a balanced subnetwork.

We now show that positive dimensional $\mathbb{C}^{*}$-zero sets always exist
for unicycle networks of homogeneous oscillators that contain a balanced
subnetwork.

######  Proposition 5.2 (Non-isolated $\mathbb{C}^{*}$-zero set for unicycle
networks).

Let $G$ be a (connected) unicycle graph on $n+1$ nodes that contains a unique
even cycle $O$. Suppose the coupling coefficients satisfy:

  1. 1.

$(G,K)$ contains a balanced subnetwork $(\vec{H},K)$;

  2. 2.

$\boldsymbol{k}^{2}(O)=c^{2}\cdot\boldsymbol{1}$ for some
$c\in\mathbb{C}^{*}$, where $\boldsymbol{k}^{2}$ the element-wise square of
$\boldsymbol{k}(O)$.

Then for any $w\in\mathbb{C}$, the algebraic Kuramoto system $\vec{f}_{G}$
derived from homogeneous network $(G,K,w\,\vec{1})$ has a positive-dimensional
$\mathbb{C}^{*}$-zero set. Moreover, after relabeling the nodes such that
$\mathcal{V}(O)=\\{0,1,\ldots,m\\}$, the $\mathbb{C}^{*}$-zero set of
$\vec{f}_{G}$ contains one-dimensional components parametrized by

$\displaystyle\begin{bmatrix}x_{1}(\lambda)&\cdots&x_{m}(\lambda)\end{bmatrix}$
$\displaystyle=(\lambda\cdot\boldsymbol{\eta}_{\vec{T}}\circ(\boldsymbol{a}(\vec{T})^{-I}))^{\check{Q}(\vec{T})^{-1}}$
$\displaystyle x_{i}$ $\displaystyle=\pm x_{\pi(i)}\quad\text{for
}i=m+1,\ldots,n,$

where $\vec{T}<\vec{H}$ such that its corresponding graph $T$ is a spanning
tree of $O$, $[\boldsymbol{\eta}_{\vec{T}}\;-1]$ is a primitive cycle vector
of $\vec{H}$, and $\pi(i)$ is the tree-order parent of node $i$.

Here, the choice $w\,\vec{1}$ for the coupling coefficients simply means that
all natural frequencies $w_{0},w_{1},\ldots,w_{n}$ are identical, i.e., the
network consists of homogeneous oscillators. Since the constant terms in the
algebraic Kuramoto system (3) are their deviations
$\overline{w}_{i}=w_{i}-\overline{w}$ from the mean, this corresponds to the
requirement that all constant terms are zero. The tree-order parent node
$\pi(i)$ of a node $i$ outside the cycle $O$ is its adjacent node in the
unique path to the reference node $0$.

######  Proof 5.3.

By recursive applications of Lemma 2.1, $\mathbb{C}^{*}$-zeros for
$\vec{f}_{O}$ always extend to $\mathbb{C}^{*}$-zeros for $\vec{f}_{G}$. So
without loss, we can assume $G=O,n=m$. Let $F=\check{\nabla}_{\vec{H}}$, then
$F$ is a facet of $\check{\nabla}_{G}$, and,

(17)
$\vec{f}_{G}(\boldsymbol{x})=\check{Q}(\vec{H})(\boldsymbol{x}^{\check{Q}(\vec{H})}\circ\boldsymbol{a}(\vec{H}))^{\top}-\check{Q}(\vec{H})(\boldsymbol{x}^{-\check{Q}(\vec{H})}\circ\boldsymbol{a}(\vec{H}))^{\top}$

Let $T$ be a spanning tree of $O=G$ and let $\vec{T}$ be the corresponding
subdigraph of $\vec{H}$. From the proof of Theorem 4.7, $\vec{T}$ gives rise
to a cocharacter $\boldsymbol{\mu}=\boldsymbol{1}\check{Q}(\vec{T})^{-1}$. It
provides the parametrization

$\boldsymbol{x}(\lambda)=(\lambda\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}}=(\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}}\circ\lambda^{\boldsymbol{\mu}}$

of a 1-dimensional orbit in the $\mathbb{C}^{*}$-zero set of
$\operatorname{init}_{F}(\vec{f}^{*}_{G})$ and therefore
$\check{Q}(\vec{H})(\boldsymbol{x}(\lambda)^{\check{Q}(\vec{H}})\circ\boldsymbol{a}(\vec{H}))^{\top}=\vec{0}$.
That is, the first term of (17) vanishes. With a straightforward calculation,
we verify

$\displaystyle\boldsymbol{x}(\lambda)^{-\check{Q}(\vec{H})}\circ\boldsymbol{a}(\vec{H})$
$\displaystyle=((\lambda\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}})^{\begin{bmatrix}-\check{Q}(\vec{T})&-\check{Q}(\vec{T})\vec{\eta}_{\vec{T}}\end{bmatrix}}\circ\boldsymbol{a}(\vec{H})=-\frac{c^{2}}{4\lambda}\cdot\boldsymbol{\eta},$

where $c$ is the constant such that
$\boldsymbol{k}^{2}(\vec{H})=c^{2}\cdot\boldsymbol{1}$ from the assumption.
Therefore,

$\check{Q}(\vec{H})(\boldsymbol{x}(\lambda)^{-\check{Q}(\vec{H})}\circ\boldsymbol{a}(\vec{H}))^{\top}=\vec{0}.$

Combined with the calculation above, we see that
$\vec{f}_{G}(\boldsymbol{x}(\lambda))=\vec{0}$ for all
$\lambda\in\mathbb{C}^{*}$ and thus the $\mathbb{C}^{*}$-zero set of
$\vec{f}_{G}=0$ is positive-dimensional.

The $\mathbb{C}^{*}$-orbit constructed here produces a positive-dimensional
real zero set to the original transcendental Kuramoto system.

######  Proposition 5.4 (Non-isolated real zero set for unicycle networks).

Let $G$ be a (connected) unicycle graph that contains a unique even cycle $O$.
Suppose the choice of the coupling coefficients satisfies

  1. 1.

$(G,K)$ contains a balanced subnetwork $(\vec{H},K)$;

  2. 2.

$\boldsymbol{k}^{2}(O)=c^{2}\cdot\boldsymbol{1}$ for some
$c\in\mathbb{R}^{+}$.

Then for any $w\in\mathbb{R}$, transcendental Kuramoto system (2) derived from
homogeneous network $(G,K,w\,\vec{1})$ has positive-dimensional real zero
sets.

######  Proof 5.5.

Let $\vec{T}$ and $\boldsymbol{\eta}_{\vec{T}}$ be as defined in the above
proof. Then for any $\lambda\in\mathbb{R}^{*}$,

$\frac{ce^{\mathfrak{i}\lambda}}{2\mathfrak{i}}\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I}=\frac{ce^{\mathfrak{i}\lambda}}{2\mathfrak{i}}\cdot\boldsymbol{\eta}_{\vec{T}}\circ\left(\frac{1}{2\mathfrak{i}}\boldsymbol{k}(\vec{T})\right)^{-I}\in(S^{1})^{m}.$

Since the group automorphism
$\boldsymbol{x}\mapsto\boldsymbol{x}^{\check{Q}(\vec{T})^{-1}}$ preserves the
the real torus $(S^{1})^{m}\subset(\mathbb{C}^{*})^{m}$, the restriction of
the parametrized zero set $\boldsymbol{x}$, defined as

$\boldsymbol{x}(\lambda)=\left(\frac{ce^{\mathfrak{i}\lambda}}{2\mathfrak{i}}\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I}\right)^{\check{Q}(\vec{T})^{-1}}\in(S^{1})^{m}$

for any $\lambda\in\mathbb{R}$. By Proposition 5.2 $\boldsymbol{x}(\lambda)$
also satisfies the algebraic Kuramoto equations. Therefore,
$\boldsymbol{\theta}(\lambda)=\log(\boldsymbol{x}(\lambda))$ is a positive-
dimensional real zero set of the transcendental Kuramoto system.

######  Example 5.6 (4-cycle, again).

For $G=C_{4}$, as noted in Example 4.24, there can be as many as 6 balanced
subnetworks, depending on the choices of coupling coefficients. They come in 3
transpose-pairs whose representatives are shown in Figure 2. Each transpose-
pair of balanced subnetworks produces an one-dimensional $\mathbb{C}^{*}$-zero
sets of $\vec{f}_{G}=0$ through the formula given in Proposition 5.2.

  1. 1.

Consider the first subdigraph $\vec{H}_{1}$ in Figure 2. It is balanced if
$k_{10}k_{32}=k_{12}k_{30}$. Then with the choice of $\vec{T}_{1}$ having arcs
$(1,0),(1,2),(3,0)$, we can compute

$\displaystyle\check{Q}(\vec{T}_{1})$
$\displaystyle=\left[\begin{smallmatrix}+1&+1&0\\\ 0&-1&0\\\
0&0&+1\end{smallmatrix}\right]$ $\displaystyle\check{Q}(\vec{T}_{1})^{-1}$
$\displaystyle=\left[\begin{smallmatrix}+1&+1&0\\\ 0&-1&0\\\
0&0&+1\end{smallmatrix}\right]$ $\displaystyle\boldsymbol{\eta}_{\vec{T}_{1}}$
$\displaystyle=\begin{bmatrix}-1&+1&+1\end{bmatrix},$

where the notation is as in Proposition 5.2. Then the function
$\boldsymbol{x}:\mathbb{C}^{*}\to(\mathbb{C}^{*})^{3}$, given by

$\displaystyle\boldsymbol{x}(\lambda)$
$\displaystyle=(\lambda\cdot\boldsymbol{\eta}_{\vec{T}_{1}}\circ\begin{bmatrix}a_{10}&a_{12}&a_{30}\end{bmatrix}^{-I})^{\check{Q}(\vec{T}_{1})^{-1}}=\begin{bmatrix}-\frac{2\mathfrak{i}\lambda}{k_{10}}&-\frac{k_{12}}{k_{10}}&\frac{2\mathfrak{i}\lambda}{k_{30}}\end{bmatrix}$

has image inside the $\mathbb{C}^{*}$-zero set of $\vec{f}_{C_{4}}$. In other
words, the function

$\boldsymbol{x}(\lambda)=\begin{bmatrix}-2\mathfrak{i}\lambda/k_{10}&-k_{12}/k_{10}&+2\mathfrak{i}\lambda/k_{30}\end{bmatrix}\quad\quad\text{if}\quad
k_{10}k_{32}=k_{12}k_{30}$

parametrizes the open part of an one-dimensional orbit in the
$\mathbb{C}^{*}$-zero set of $\vec{f}_{C_{4}}$. In the special case when
$k_{ij}=1$, this one-dimensional orbit is _balanced_ in the sense of [32,
Definition 3.1] and [32] provides analysis of the stability and geometry of
such orbits.

  2. 2.

The second subnetwork $\vec{H}_{2}$ in Figure 2 is balanced if
$k_{10}k_{30}=k_{12}k_{23}$. With $\vec{T}_{2}<\vec{H}_{2}$ given by arcs
$(1,0),(1,2),(0,3)$ and by following the construction above, we have

$\boldsymbol{x}(\lambda)=\begin{bmatrix}+2\mathfrak{i}\lambda/k_{10}&-k_{12}/k_{10}&+k_{03}/2\mathfrak{i}\lambda\end{bmatrix}\quad\quad\text{if}\quad
k_{10}k_{30}=k_{12}k_{23}$

whose image is in the $\mathbb{C}^{*}$-zero set of $\vec{f}_{C_{4}}$ for any
$\lambda\in\mathbb{C}^{*}$. Note that even in the special case of $k_{ij}=1$,
this orbit is _not_ balanced (in the sense of [32, Definition 3.1]) for
$\lambda\neq 1$.

  3. 3.

Finally, the third subnetwork $\vec{H}_{3}$ in Figure 2 is balanced if
$k_{10}k_{12}=k_{32}k_{03}$. With the choice $\vec{T}_{3}<\vec{H}_{3}$, and
following the construction above, we have

$\boldsymbol{x}(\lambda)=\begin{bmatrix}+k_{01}/2\mathfrak{i}\lambda&-k_{01}k_{12}/4\lambda^{2}&-k_{03}/2\mathfrak{i}\lambda\end{bmatrix}\quad\text{if}\quad
k_{10}k_{12}=k_{32}k_{30}$

whose image is in the $\mathbb{C}^{*}$-zero set of $\vec{f}_{C_{4}}$ for any
$\lambda\in\mathbb{C}^{*}$. This orbit is more interesting as all three
complex phase variables $x_{1},x_{2},x_{3}$ are nonconstant relative to the
reference phase $x_{0}=1$. It is also _not_ balanced (in the sense of [32,
Definition 3.1]).

Moreover, if the coupling coefficients satisfy the condition
$\boldsymbol{k}^{2}(C_{4})=c^{2}\cdot\boldsymbol{1}$ for some
$c\in\mathbb{R}$, then the three $\mathbb{C}^{*}$-orbits contain one-real-
dimensional components on the real torus $(S^{1})^{3}$. Indeed, by the
restriction $\lambda=\frac{ce^{\mathfrak{i}t}}{2\mathfrak{i}}$, the three
orbits constructed above reduce to parametrized zero sets of one real-
dimension inside the real torus. Here, $\operatorname{sgn}(x)\in\\{\pm 1\\}$
denotes the sign of $x\in\mathbb{R}$. If we define

$\sigma^{ij}_{\ell m}=\begin{cases}0&\text{if }k_{ij}/k_{\ell m}>0\\\
\pi&\text{otherwise},\end{cases}\quad\text{and}\quad\sigma^{ij}=\begin{cases}0&\text{if
}k_{ij}>0\\\ \pi&\text{otherwise},\end{cases}$

then the three potential real orbits can be expressed as

(18) $\displaystyle\left\\{\begin{aligned} \theta_{1}&=t+\pi+\sigma^{10}\\\
\theta_{2}&=\pi+\sigma^{12}_{10}\\\ \theta_{3}&=t+\sigma^{30}\\\
\end{aligned}\right.\text{if }\frac{k_{10}k_{32}}{k_{12}k_{30}}=1,$
$\displaystyle\left\\{\begin{aligned} \theta_{1}&=\sigma^{10}+t\\\
\theta_{2}&=\pi+\sigma^{21}_{10}\\\
\theta_{3}&=\sigma^{30}-t\end{aligned}\right.\text{if
}\frac{k_{10}k_{30}}{k_{12}k_{23}}=1,$ $\displaystyle\left\\{\begin{aligned}
\theta_{1}&=\sigma^{10}-t\\\ \theta_{2}&=\sigma^{21}_{10}-2t\\\
\theta_{3}&=\pi+\sigma^{30}-t\\\ \end{aligned}\right.\text{if
}\frac{k_{10}k_{12}}{k_{32}k_{30}}=1,$

with $\theta_{0}=0$. It is easy to see if $k_{ij}$’s are identical, then all
three real orbits exist. Moreover, these three orbits intersects at a singular
point $(\theta_{0},\theta_{1},\theta_{2},\theta_{3})=(0,\pi/2,\pi,\pi/2)$.
These positive-dimensional solution sets have been studied in Refs. [28, 32]
([32, Section 5.2], in particular, also provide topological analysis for the
orbits). In this example, we showed they can also be derived systematically
from balanced subnetworks.

###  5.2 Multiple even cycles sharing one edge

We now show positive-dimensional zero sets for $\vec{f}_{G}$ is possible for
networks consisting of multiple even cycles sharing a single edge (e.g.,
Figure 3(a)).

######  Proposition 5.7.

Suppose $G$ consists of $d$ independent even cycles $C_{1},\ldots,C_{d}$ that
share a single edge $e$, then with the choice of the coupling coefficients

$k_{ij}=\begin{cases}sd&\text{if }\\{i,j\\}=e\\\
s&\text{othewise},\end{cases}$

for any $s\in\mathbb{C}^{*}$, and homogeneous natural frequencies
$\vec{w}=w\cdot\vec{1}$ for any $w\in\mathbb{C}^{*}$, the algebraic Kuramoto
system $\vec{f}_{G}$ derived from the network $(G,K,w\cdot\vec{1})$ has a
positive-dimensional $\mathbb{C}^{*}$-zero set.

######  Proof 5.8.

Since $G$ is bipartite, by Theorem 2.3, there exists a facet $F$ of
$\check{\nabla}_{G}$ such that $G_{F}=G$, and thus
$\check{\nabla}_{G}=\\{\vec{0}\\}\cup F\cup(-F)$. Consequently,

$\vec{f}_{G}(\boldsymbol{x})=\check{Q}(\vec{G}_{F})(\boldsymbol{x}(\lambda)^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F}))^{\top}+\check{Q}(\vec{G}_{-F})(\boldsymbol{x}(\lambda)^{\check{Q}(\vec{G}_{-F})}\circ\boldsymbol{a}(\vec{G}_{-F}))^{\top}.$

Let $T$ be a spanning tree of $G$ that contains $e$, and let $\vec{T}$ be the
corresponding subdigraph such that $\vec{T}<\vec{G}_{F}$. We shall arrange of
the columns of $\check{Q}(\vec{G}_{F})$ and $\check{Q}(\vec{T})$ so that

$\displaystyle\check{Q}(\vec{G}_{F})$
$\displaystyle=\begin{bmatrix}\check{Q}(\vec{T})&\vec{v}_{1}&\cdots&\vec{v}_{d}\end{bmatrix}$
and $\displaystyle\check{Q}(\vec{T})$
$\displaystyle=\begin{bmatrix}\check{Q}(\vec{T}-\\{e\\})&\vec{v}_{e}\end{bmatrix}$

where $\vec{v}_{e}$ is the incidence vector corresponding to the edge $e$ that
is shared by all cycles, and $\vec{v}_{1},\ldots,\vec{v}_{d}$ are the
incidence vectors corresponding to the arcs associated with the edges
$e_{1},\ldots,e_{d}$ in
$\mathcal{E}(C_{1})\setminus\mathcal{E}(T),\ldots,\mathcal{E}(C_{d})\setminus\mathcal{E}(T)$,
respectively. Pick basis vectors $\vec{\eta}_{1},\ldots,\vec{\eta}_{d}$ of
$\ker\check{Q}(\vec{G}_{F})$ that are primitive cycle vector associated with
the cycles $C_{1},\ldots,C_{d}$ with consistent signs (i.e., their shared
nonzero entries have the same sign). Let

$\boldsymbol{\eta}=\begin{bmatrix}\boldsymbol{\eta}_{T}&u_{1}&\cdots&u_{d}\end{bmatrix}:=(\vec{\eta}_{1}+\cdots+\vec{\eta}_{d})^{\top}.$

Then the first $n-1$ entries of $\boldsymbol{\eta}_{T}$ are $\pm 1$, and its
$n$-th entry is $\pm d$. The signs depend on the choices of the primitive
cycle vectors.

With these, we define the nonconstant function
$\boldsymbol{x}:\mathbb{C}^{*}\to(\mathbb{C}^{*})^{n}$, given by

$\boldsymbol{x}(\lambda)=(\lambda\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}},$

as above. Then, as demonstrated in Theorem 4.7,

$\check{Q}(\vec{G}_{F})(\boldsymbol{x}(\lambda)^{\check{Q}(\vec{G}_{F})}\circ\boldsymbol{a}(\vec{G}_{F}))^{\top}=\vec{0}$

for any $\lambda\in\mathbb{C}^{*}$. It remains to show
$\check{Q}(\vec{G}_{-F})(\boldsymbol{x}(\lambda)^{\check{Q}(\vec{G}_{-F})}\circ\boldsymbol{a}(\vec{G}_{-F}))^{\top}$
also vanishes. Since
$\boldsymbol{a}(\vec{T})=\frac{\boldsymbol{k}(\vec{T})}{2\mathfrak{i}}=[\;s\;\cdots\;s\;sd\;]$,

$\displaystyle\boldsymbol{x}(\lambda)^{-\check{Q}(\vec{T})}\circ\boldsymbol{a}(\vec{T})$
$\displaystyle=(\lambda\cdot\boldsymbol{\eta}_{\vec{T}}\circ\boldsymbol{a}(\vec{T})^{-I})^{-I}\circ\boldsymbol{a}(\vec{T})=\lambda^{-1}\boldsymbol{\eta}_{\vec{T}}^{-I}\circ\begin{bmatrix}\frac{s^{2}}{-4}&\cdots&\frac{s^{2}}{-4}&\frac{s^{2}d^{2}}{-4}\end{bmatrix}=\frac{s^{2}}{-4\lambda}\,\boldsymbol{\eta}_{\vec{T}}$

Moreover, by construction,
$\boldsymbol{x}(\lambda)^{\vec{v}_{i}}\cdot\boldsymbol{a}(e_{i})=\lambda
u_{i}$ for $i=1,\ldots,d$. Therefore,

$\displaystyle\boldsymbol{x}(\lambda)^{-\vec{v}_{i}}\cdot\boldsymbol{a}(e_{i})$
$\displaystyle=\lambda^{-1}\boldsymbol{a}^{2}(e_{i})u_{i}^{-1}=\frac{s^{2}}{-4\lambda}\,u_{i},$

since $u_{1}\in\\{\pm 1\\}$. This shows that

$\displaystyle\boldsymbol{x}(\lambda)^{\check{Q}(\vec{G}_{-F})}\circ\boldsymbol{a}(\vec{G}_{-F})$
$\displaystyle=\begin{bmatrix}\boldsymbol{x}(\lambda)^{-\check{Q}(\vec{T})}\circ\boldsymbol{a}(\vec{T})&\boldsymbol{x}(\lambda)^{-\vec{v}_{1}}\cdot\boldsymbol{a}(e_{1})&\boldsymbol{x}(\lambda)^{-\vec{v}_{2}}\cdot\boldsymbol{a}(e_{1})\end{bmatrix}=\frac{s^{2}}{-4\lambda}\,\boldsymbol{\eta}.$

Consequently, $\vec{f}_{G}(\boldsymbol{x}(\lambda))=\vec{0}$ for any
$\lambda\in\mathbb{C}^{*}$, i.e., the $\mathbb{C}^{*}$-zero set of
$\vec{f}_{G}$ is positive-dimensional.

$0$$1$$2$$3$$4$$5$$s$$s$$s$$s$$s$$s$$2s$ (a) $G$

$0$$1$$2$$3$$4$$5$ (b) A facet subdigraph $\vec{G}_{F}$

$0$$1$$2$$3$$4$$5$ (c) $\vec{T}$ inside $\vec{G}_{F}$

Figure 3:  A network with two independent even cycles.

######  Example 5.9.

Consider the network shown in Figure 3(a). We fix a particular facet whose
facet subdigraph is shown in Figure 3(b). We also fix a choice of
$\vec{T}<\vec{G}_{F}$ shown in Figure 3(c). With this choice of $\vec{T}$ and
the ordering of the arcs $(1,0)$, $(1,2)$, $(3,2)$, $(5,0)$, $(1,4)$, $(3,4)$,
$(5,4)$, we have

$\displaystyle\check{Q}(\vec{T})$
$\displaystyle=\left[\begin{smallmatrix}1&1&0&0&1\\\ 0&-1&-1&0&0\\\
0&0&1&0&0\\\ 0&0&0&0&-1\\\ 0&0&0&1&0\end{smallmatrix}\right],$
$\displaystyle\check{Q}(\vec{T})^{-1}$
$\displaystyle=\left[\begin{smallmatrix}1&1&1&1&0\\\ 0&-1&-1&0&0\\\
0&0&1&0&0\\\ 0&0&0&0&1\\\ 0&0&0&-1&0\end{smallmatrix}\right],$
$\displaystyle\vec{\eta}_{1}$ $\displaystyle=\left[\begin{smallmatrix}0\\\
-1\\\ +1\\\ 0\\\ +1\\\ -1\\\ 0\end{smallmatrix}\right],$
$\displaystyle\vec{\eta}_{2}$ $\displaystyle=\left[\begin{smallmatrix}-1\\\
0\\\ 0\\\ +1\\\ +1\\\ 0\\\ -1\end{smallmatrix}\right]$

as constructed in the proof above. We also define

$\displaystyle\boldsymbol{\eta}$
$\displaystyle=\vec{\eta}_{1}^{\top}+\vec{\eta}_{2}^{\top}=\begin{bmatrix}-1&-1&+1&+1&+2&-1&-1\end{bmatrix}$
and $\displaystyle\boldsymbol{\eta}_{T}$
$\displaystyle=\begin{bmatrix}-1&-1&+1&+1&+2\end{bmatrix}.$

We then verify that the construction
$\boldsymbol{x}(\lambda)=(\lambda\cdot\boldsymbol{\eta}_{T}\circ\boldsymbol{a}(\vec{T})^{-I})^{\check{Q}(\vec{T})^{-1}}$
above produces

$\displaystyle(\lambda\begin{bmatrix}-1&-1&+1&+1&+2\end{bmatrix}\circ\begin{bmatrix}\frac{s}{2\mathfrak{i}}&\frac{s}{2\mathfrak{i}}&\frac{s}{2\mathfrak{i}}&\frac{s}{2\mathfrak{i}}&\frac{2s}{2\mathfrak{i}}\end{bmatrix}^{-I})^{\check{Q}(\vec{T})^{-1}}$
$\displaystyle=\begin{bmatrix}-\frac{2\mathfrak{i}\lambda}{s}&+1&+\frac{2\mathfrak{i}\lambda}{s}&-1&+\frac{2\mathfrak{i}\lambda}{s}\end{bmatrix},$

which parametrizes an one-dimensional $\mathbb{C}^{*}$-zero set of
$\vec{f}_{G}=0$. Moreover, by choosing
$\lambda(t)=\frac{se^{\mathfrak{i}t}}{2\mathfrak{i}}$,
$\boldsymbol{x}\in(S^{1})^{5}$ for any $t\in\mathbb{R}$. Therefore,
$\log(\boldsymbol{x})$ produces the one-dimensional real zero set

$(\theta_{0},\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})=(0,t+\pi,0,t,\pi,t)$

for the transcendental Kuramoto system (2) derived from the network in Figure
3(a).

##  6 Concluding remarks

We studied the structure of the zero sets of the Kuramoto equations and its
algebraic counterpart. By leveraging a recently discovered three-way
connection between the graph-theoretic, convex-geometric, and tropical view
points, we answered four key questions.

  1. 1.

For Question 1.1, we showed that _for generic natural frequencies and generic
but symmetric coupling coefficients, the $\mathbb{C}^{*}$ root count for the
algebraic Kuramoto system coincides with the adjacency polytope bound, and as
a corollary we showed that the algebraic Kuramoto system is Bernshtein-
general. _

  2. 2.

For Question 1.2, we provided a description of the _exceptional coupling
coefficients for which the $\mathbb{C}^{*}$ root count of the algebraic
Kuramoto system drops below the generic zero count. _ This description used
graph-theoretic, combinatorial and toric information.

  3. 3.

For Question 1.3, we analyzed unicycle networks and developed a _full
stratification of the coupling coefficient space and computed the
$\mathbb{C}^{*}$ root count over each stratum. _

  4. 4.

For Question 1.4, we established sufficient conditions on unicycle networks
and networks consisting of cycles sharing a single edge under which _there
will be non-isolated real and complex zero sets for Kuramoto systems_ through
explicit constructions.

While the analysis for the last two questions required some topological
restrictions, it appears hopeful that the approach taken here can be
generalized to other networks. We hope our work will spark interest in the
full analysis of the typical and atypical solutions to Kuramoto equations.

The proofs are constructive. In particular, the homotopies used in Theorem 3.7
and Lemma 4.21 are specialized polyhedral homotopy [23] that will likely bring
significant performance improvement.

Finally, we speculate that the explicit connection between the root count of
the algebraic Kuramoto equations and the normalized volume of adjacency
polytopes may also allow algebraic geometers to directly contribute to the
geometric study of adjacency polytopes.

## Acknowledgments

This project is inspired by a series of discussion the first named author had
with Anton Leykin, Josephine Yu, and Yue Ren between 2017 and 2018. The first
named author also learned much about the structure of adjacency polytopes
(a.k.a. symmetric edge polytopes) from Robert Davis, Alessio D’Alì, Emanuele
Delucchi, and Mateusz Michałek. The authors thank Paul Breiding, Paul
Helminck, and Davide Sclosa for their comments on an earlier version of this
manuscript.

##  Appendix A Notations

Here, we list notations used in this paper that may not be standard.

$\mathbb{C}\\{\tau\\}$

The field of Puiseux series in $\tau$ with complex coefficients is denoted
$\mathbb{C}\\{\tau\\}$, and only convergent series, representing germs of one-
dimensional analytic varieties, will be relevant.

$\mathcal{C}(P,\Delta_{\omega})$

The (closed) secondary cone of a regular subdivision $\Delta_{\omega}$ in a
polytope $P$.

$\Sigma_{\omega}(P)$

The regular subdivision of a point configuration $P$ induced by a lifting
function $\omega:P\to\mathbb{Q}$.

$\check{\nabla}_{G}$

The point configuration associated with the adjacency polytope derived from a
connected graph $G$. It is defined to be
$\\{\pm(\vec{e}_{i}-\vec{e}_{j})\mid\\{i,j\\}\in\mathcal{E}(G)\\}\cup\\{\vec{0}\\}$.

$\check{\nabla}_{G}^{\omega}$

“Lifted” version of point configuration $\check{\nabla}_{G}$ induced by a
lifting function $\omega:\check{\nabla}_{G}\to\mathbb{Q}$. It consists of the
points $(\vec{p},\omega(\vec{p}))\subset\mathbb{R}^{n+1}$ for all
$\vec{p}\in\check{\nabla}_{G}$.

$\vec{f}_{(G,K,\vec{w})},\vec{f}_{(G,K)},\vec{f}_{G}$

The algebraic Kuramoto system derived from a Kuramoto network $(G,K,\vec{w})$.
If the choice of the natural frequencies $\vec{w}$ is not relevant to the
discussions (or assumed to be generic), the notation $\vec{f}_{(G,K)}$ will be
used. Similarly, if only graph topology is of relevance, we will simply use
$\vec{f}_{G}$.

$\vec{f}^{*}_{G}$

The randomized algebraic Kuramoto system, i.e., $R\cdot\vec{f}_{G}$ for a
generic square matrix $R$.

$(G,K,\vec{w}),(G,K)$