we shall reformulate the synchronization system  as a system of rational equations. Using the identity $\sin(\theta_{i} - \theta_{j}) = \frac{1}{2{\mathbf{i}}}
(e^{ {\mathbf{i}}(\theta_{i} - \theta_{j})} - e^{-{\mathbf{i}}(\theta_{i} - \theta_{j})})$ where ${\mathbf{i}}= \sqrt{-1}$,  can be transformed into $$\omega_{i} -
    \sum_{j \in \mathcal{N}_G(i)}
    \frac{a_{i,j}}{2{\mathbf{i}}} (
    e^{ {\mathbf{i}}\theta_{i}} e^{-{\mathbf{i}}\theta_{j}} -
    e^{-{\mathbf{i}}\theta_{i}} e^{ {\mathbf{i}}\theta_{j}}
    ) = 0
    \quad \text{ for } i = 1,\dots,n.$$ With the substitution $x_{i} := e^{{\mathbf{i}}\theta_{i}}$ for $i = 1,\dots,n$, we obtain the Laurent polynomial system $$\label{equ:sync-laurent}
    F_{G,i}(x_1,\dots,x_n) = \omega_{i} - \sum_{j \in \mathcal{N}_G(i)} a_{ij}'
    \left(
        \frac{x_i}{x_j} - \frac{x_j}{x_i}
    \right) = 0
    \quad \text{ for } i = 1,\dots,n$$ where $a_{ij}' = \frac{a_{ij}}{2{\mathbf{i}}}$ and $x_0 = 1$ is a constant. This system, $F_G = (F_{G,1},\dots,F_{G,n})$, is a system of $n$ rational equations in the $n$ complex variables $\mathbf{x} = (x_1,\dots,x_n)$. Since $x_i$’s appear in the denominator positions, $F_G$ is only defined on ${({\mathbb{C}}^*)}^n = {({\mathbb{C}}\setminus \{0\})}^n$. Clearly, each equivalence class of real solutions of  (modulo translations by multiples of $2\pi$) corresponds to a solution of  in ${({\mathbb{C}}^*)}^n$. Therefore, we can consider a more general root counting problem:

\[prb:Cstar\] Given nonzero constants $\omega_1,\dots,\omega_n$ and a weighted graph of $n+1$ nodes with weights $\{ a_{ij}' \}$, what is the maximum number of isolated 