Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

TL;DR

This paper extends Newton polyhedron theory to compute discrete invariants of algebraic varieties defined by generically inconsistent systems of Laurent polynomials, incorporating supports and polyhedra into the analysis.

Contribution

It introduces a method to analyze systems that are generically inconsistent, linking supports and Newton polyhedra to invariants of the solution set.

Findings

Computed invariants for consistent systems with fixed supports.

Extended Newton polyhedron theory to inconsistent systems.

Revealed the role of supports in the invariants of algebraic varieties.

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.