Equilibria of the rank-one Kuramoto model]{The maximum number of equilibria of the Kuramoto model with rank-one coupling

The Kuramoto model describes the phases of coupled oscillators, each with its own natural frequency. In the rank-one model, each pair of oscillators is coupled with a strength proportional to the product of the two oscillators' positive weights, so the coupling matrix has rank one. The equilibria of the rank-one model, the configurations in which every phase stays constant, are counted up to a common shift of all phases and are the real solutions of a system of polynomial equations in the sines and cosines of the phases. Coss, Hauenstein, Hong and Molzahn bounded the number of equilibria by the number of complex solutions that this system has for generic weights and natural frequencies. They obtained exact counts for equal weights and natural frequencies that take two opposite values, or two opposite values and zero. They conjectured that the largest of these counts, which is smaller than the bound from four oscillators on, is the maximum number of equilibria over all weights and natural frequencies. The bound is, however, attained for every number of oscillators from two on, so the conjecture fails from four oscillators on. The proof counts the equilibria whenever the natural frequencies have magnitudes proportional to the weights. The bound is then attained when the weights are proportional to numbers that double from one oscillator to the next, starting from one, except that the last number is one less than twice the number before it, and when the natural frequencies are small and sum to zero.