Gale duality for complete intersections

TL;DR

This paper establishes a duality between complete intersections of Laurent polynomials and master functions, enabling new methods to analyze solutions and topological properties of these systems.

Contribution

It introduces Gale duality as a correspondence between Laurent polynomial intersections and master function intersections, providing new tools for their study.

Findings

Gale duality links Laurent polynomial systems with master functions.

A Kouchnirenko-type theorem for master functions is proved.

Topological invariants of generic master function intersections are computed.

Abstract

We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections.