Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

Abstract

Let ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_k$ be finite sets in ${\mathbb{Z}}^n$ and let $Y\subset ({\mathbb{C}}^{\ast }{)}^n$ be an algebraic variety defined by a system of equations \[ f_1 = \ldots = f_k = 0, \] where $f_1,\dots ,f_k$ are Laurent polynomials with supports in ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_k$. Assuming that $f_1,\dots ,f_k$ are sufficiently generic, the Newton polyhedron theory computes discrete invariants of $Y$ in terms of the Newton polyhedra of $f_1,\dots ,f_k$. It may appear that the generic system with fixed supports ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_k$ 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 ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_k$ 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 ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_k$ themselves appear in the answers.

10 pages, subsection 4.1 is added, comments are welcome