  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.
