Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

Let $\mathcal{A}_1, \ldots, \mathcal{A}_k$ be finite sets in $\mathbb{Z}^n$ and let $Y \subset (\mathbb{C}^*)^n$ be an algebraic variety defined by a system of equations $$f_1 = \ldots = f_k = 0,$$ where $f_1, \ldots, f_k$ are Laurent polynomials with supports in $\mathcal{A}_1, \ldots, \mathcal{A}_k$. Assuming that $f_1, \ldots, f_k$ are sufficiently generic, the Newton polyhedron theory computes discrete invariants of $Y$ in terms of the Newton polyhedra of $f_1, \ldots, f_k$. It may appear that the generic system with fixed supports $\mathcal{A}_1, \ldots, \mathcal{A}_k$ is inconsistent. In this paper, we compute discrete invariants of algebraic varieties defined by system of equations which are generic in the set of consistent system with support in $\mathcal{A}_1, \ldots, \mathcal{A}_k$ by reducing the question to the Newton polyhedra theory. Unlike the classical situation, not only the Newton polyhedra of $f_1,\dots,f_k$, but also the supports $\mathcal{A}_1, \ldots, \mathcal{A}_k$ themselves appear in the answers.