---
abstract: 'Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.'
author:
- 'Tianran Chen [^1]'
- 'Robert Davis [^2]'
- 'Dhagash Mehta [^3]'
bibliography:
- 'nobodies.bib'
- 'kuramoto.bib'
- 'conv.bib'
- 'bkk.bib'
- 'chen.bib'
title: Counting equilibria of the Kuramoto model using birationally invariant intersection index
---

Introduction {#sec:introduction}
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The root counting problem for systems of nonlinear equations is a fundamental problem in mathematics that has a wide range of applications. Given an algebraic variety $X$ and complex vector spaces $L_1,\dots,L_n$ of rational functions on $X$, it has been established by K. Kaveh and A.G. Khovanskii [@kaveh_newton_2009] that for generic choices $f_1 \in L_1,\dots,f_n \in L_n$, the number of common complex roots of $f_1,\dots,f_n$ in $X$ is a fixed number, 