Irreducibility of toric complete intersections

TL;DR

This paper introduces a new combinatorial approach to determine the irreducibility of generic complete intersections in algebraic tori, extending existing theorems to arbitrary characteristic fields and providing practical criteria for specific cases.

Contribution

It generalizes Khovanskii's irreducibility theorems to all characteristics and offers combinatorial conditions for irreducibility of engineered complete intersections.

Findings

Extended irreducibility theorems to arbitrary characteristic fields.

Provided combinatorial criteria for irreducibility of certain critical loci.

Developed an approach applicable to fixed monomials and linear relations on coefficients.

Abstract

We develop an approach to study the irreducibility of generic complete intersections in the algebraic torus defined by equations with fixed monomials and fixed linear relations on coefficients. Using our approach we generalize the irreducibility theorems of Khovanskii to fields of arbitrary characteristic. Also we get a combinatorial sufficient conditions for irreducibility of engineered complete intersections. As an application we give a combinatorial condition of irreducibility for some critical loci and Thom-Bordmann strata: $f = f'_x = 0$, $f'_x = f'_y = 0$, $f = f'_x = f'_{xx} = 0$, where f is a generic Laurent polynomial with a prescribed monomial set.