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The Kuramoto model is a classical model used in the study of spontaneoussynchronizations in networks of coupled oscillators. In this model, frequencysynchronization configurations can be formulated as complex solutions to asystem of algebraic equations. Recently, upper bounds to the number offrequency synchronization configurations in cycle networks of N oscillatorswere calculated under the assumption of generic non-uniform coupling. In thispaper, we refine these results for the special cases of uniform coupling. Inparticular, we show that when, and only when, N is divisible by 4, the upperbound for the number of synchronization configurations in the uniform couplingcases is significantly less than the bound in the non-uniform coupling cases.This result also establishes an explicit formula for the gap between thebirationally invariant intersection index and theBernshtein-Kushnirenko-Khovanskii bound for the underlying algebraic equations.

On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling