for certain small graphs [@Chen2016; @mehta2011finding; @Mehta2015; @Mehta:2009zv]. Recently, in the special case of “rank-one coupling”, i.e., the matrix $[a_{ij}]$ has rank 1, a much smaller bound $2^N-2$ was established [@Coss2017]. Based on the theory of the BKK bound, a search for topology-dependent bounds on the number of solutions to  and  was initiated in [@Chen2016; @Mehta2015]. In the present contribution, we provide explicit formulas for a much stronger solution bound: the birationally invariant intersection index.

Preliminaries and notations {#sec:prelim}
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For a compact set $Q \subset {\mathbb{R}}^n$, $\operatorname{vol}_n(Q)$ denotes its standard Euclidean volume, and the quantity $n! \operatorname{vol}_n(Q)$ is its *normalized volume*, denoted $\operatorname{NVol}_n(Q)$. Say $Q$ is *convex* if it contains the line segment connecting any two points $Q$. For a set $X \subset {\mathbb{R}}^n$, its *convex hull* is the smallest convex set containing it, denoted $\operatorname{conv}(X)$, and its *affine span* is the smallest affine subspace of ${\mathbb{R}}^n$ containing it, denoted $\operatorname{aff}(X)$. A (*convex*) *polytope* is the convex hull of a finite set of points. Of particular importance in the current context are convex polytopes whose vertices lie in ${\mathbb{Z}}^n$. Such polytopes are called *lattice polytopes*. A full dimensional convex lattice polytope $P \subset {\mathbb{R}}^n$ is said to be *reflexive* 