  The Kuramoto model is a classical model used in the study of spontaneous
synchronizations in networks of coupled oscillators. In this model, frequency
synchronization configurations can be formulated as complex solutions to a
system of algebraic equations. Recently, upper bounds to the number of
frequency synchronization configurations in cycle networks of N oscillators
were calculated under the assumption of generic non-uniform coupling. In this
paper, we refine these results for the special cases of uniform coupling. In
particular, we show that when, and only when, N is divisible by 4, the upper
bound for the number of synchronization configurations in the uniform coupling
cases is significantly less than the bound in the non-uniform coupling cases.
This result also establishes an explicit formula for the gap between the
birationally invariant intersection index and the
Bernshtein-Kushnirenko-Khovanskii bound for the underlying algebraic equations.

Algebraic Geometry