Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

Abstract

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.