Let $T^n$ be a $n$-dimensional complex torus, $\mathbb{Z}^n$ be the lattice of its characters, $E$ be a $\mathbb{C}$-vector space with $\dim E=r$. Let $A\subset \mathbb{Z}^n$ be a finite set. Consider a map from $A$ to the set of subspaces in $E$ which assigns to $\alpha\in A$ a space $E_\alpha\subset E$. The following problem generalizes a classical problem from the Newton Polyhedra Theory:

\noindent{\bf Problem.} Let $X\subset T^n$ be the set of solutions of a following equation

$$\sum_{\alpha\in A} e_\alpha x^\alpha=0,$$

where $e_\alpha$ is a generic vector from $E_\alpha$ and $x^\alpha$ is the character corresponding to~$\alpha\in \mathbb{Z}^n$. Find discrete invariants of $X$ (like the Euler characteristic, the arithmetic genus, the $h^{p,q}_k$ numbers of the mixed Hodge structure on the cohomology group $H^k(X)$).

\smallskip

I will present a formula for the number of solutions, i.e., for the number of points in $X$, for a generic equation with $r=\dim E=n$. More generally I will present an explicit combinatorial description of $X$ as an element of the ring of conditions of $T^n$ for $\dim E \leq n$.

The talk is based on a joint paper in preparation written with K.~Kaveh and H.~Spink. Our work was inspired by the beautiful results of June Huh and by Klyachko's combinatorial description of invariant vector bundles on toric varieties.