known as the *birationally invariant intersection index* of $L_1,\dots,L_n$ in $X$, denoted $[L_1,\dots,L_n]$. Moreover, $[L_1,\dots,L_n]$ is given by the mixed volume of Newton-Okounkov bodies associated with $L_1,\dots,L_n$ and hence a far generalization of the well known BKK bound [@bernshtein_number_1975; @khovanskii_newton_1978; @kushnirenko_newton_1976]. Computation of $[L_1,\dots,L_n]$ remains difficult. This paper focuses on the indirect computation of this index for an algebraic formulation of the “Kuramoto equations” rooted in the study of spontaneous synchronization in networks of connected oscillators which is a ubiquitous phenomenon that has been discovered and studied in a wide range of disciplines including physics, biology, chemistry, and engineering [@dorfler_synchronization_2014]. Mathematically, a network of $N=n+1$ oscillators can be described by a weighted graph $G = (V,E,A)$ in which vertices $V = \{0,\dots,n\}$ represent the oscillators, edges $E$ represent their connections, and weights $A = \{a_{ij}\}$ represent the *coupling strength* along edges. In isolation, the oscillators have their own natural frequency $\omega_0,\dots,\omega_n$. However, in a network of oscillators the tug of war between the oscillators’ tendency to oscillate in their own natural frequencies and the influence of their neighbors gives rise to rich and complicated phenomenon. This is captured by the Kuramoto model [@Kuramoto1975] $$\frac{d \theta_i}{dt} =
    \omega_i -
    \sum_{j \in 