\mathcal{N}_G(i)} a_{ij} \sin(\theta_{i}-\theta_{j})
    \quad \text{ for } i = 0,\dots,n
    \label{equ:kuramoto-ode}$$ where each $\theta_i \in [0,2\pi)$ is the phase angle that describes the status of the $i$-th oscillator, and $\mathcal{N}_G(i)$ is the set of neighbors of the $i$-th vertex. A configuration $\boldsymbol{\theta} = (\theta_0,\dots,\theta_n)$ is said to be in *frequency synchronization* if $\frac{d\theta_i}{dt} = 0$ for all $i$ at $\boldsymbol{\theta}$. To remove the inherent degree of freedom given by uniform rotations, it is customary to fix $\theta_0 = 0$. Then such synchronization configurations are characterized by the system of $n$ nonlinear equations $$\omega_i - \sum_{j \in \mathcal{N}_G(i)} a_{ij} \sin(\theta_{i}-\theta_{j}) = 0
    \quad \text{ for } i = 1,\dots,n
    \label{equ:sync-sin}$$ in the variables $\theta_1,\dots,\theta_n$ with constant $\theta_0 = 0$. Then, the root counting problem is:

\[prb:original\] Given $\omega_1,\dots,\omega_n \in {\mathbb{R}}$ and a weighted graph of $n+1$ nodes, what is the maximum number of real solutions the induced system  could have?

An upper bound to this answer, that is independent from network topology, is shown to be $\binom{2n}{n}$ [@Baillieul1982]. However, recent studies [@Chen2016; @Mehta2015] suggests much tighter upper bounds that are sensitive to network topology may exist. In this paper, we show that this is true.

To leverage tools from algebraic geometry, 