Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appears 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 upper bounds on this number using the theory of the birationally invariant intersection index of a family of rational functions. These bounds are significant and make asymptotic improvements over the best existing bound.