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.