${\mathbb{C}}^*$-solutions the system  could have?

Clearly, every answer for Problem \[prb:Cstar\] would provide an upper bound for the answers for Problem \[prb:original\]. However, the algebraic formulation for Problem \[prb:Cstar\] allows the use of powerful tools from complex algebraic geometry, in particular, the theory of birationally invariant intersection index which states that the maximum number of isolated solutions coincide with the “generic” number of isolated solutions of an appropriate family of systems: For each vertex $i=1,\dots,n$, define the complex vector space of rational functions $$L_{G,i} = \operatorname{span} \left(
        \{1\} \; \cup \;
        {\{ x_i x_j^{-1} - x_i^{-1} x_j \}}_{j\in\mathcal{N}_G(i)}
        \right).
    \label{equ:L-space}$$ With this construction, the $i$-th equation in  is an element in $L_{G,i}$. Therefore, the number of ${\mathbb{C}}^*$-solutions of  for generic choices of weights and constant terms will be equal to the number of common roots of $n$ generic elements from $L_{G,1},\dots,L_{G,n}$ respectively within the toric variety ${({\mathbb{C}}^*)}^n$. This is precisely the birationally invariant intersection index [@kaveh_newton_2009], denoted $[L_{G,1},\dots,L_{G,n}]$.

\[prb:index\] Given a graph $G$ with $n+1$ vertices $0,1,\dots,n$, let $L_{G,i} = \operatorname{span} \left(
   \{1\} \; \cup \;
   {\{ x_i x_j^{-1} - x_i^{-1} x_j \}}_{j\in\mathcal{N}(i)}
   \right)$. What is $[\; L_{G,1}\,,\, \dots \, ,\, L_{G,n} \;]$ ?

Though the intersection index $[L_{G,1},\dots,L_{G,n}]$ can 