Affine solution sets of sparse polynomial systems

TL;DR

This paper studies the structure of affine varieties defined by sparse polynomial systems, providing combinatorial conditions for positive dimensional components, and introduces algorithms for their decomposition and degree bounds.

Contribution

It offers new combinatorial criteria for positive dimensional components and algorithms for equidimensional decomposition of sparse polynomial systems.

Findings

Combinatorial conditions for positive dimensional components.

Algorithms for equidimensional decomposition.

Upper bounds for the degree of affine varieties.

Abstract

This paper focuses on the equidimensional decomposition of affine varieties defined by sparse polynomial systems. For generic systems with fixed supports, we give combinatorial conditions for the existence of positive dimensional components which characterize the equidimensional decomposition of the associated affine variety. This result is applied to design an equidimensional decomposition algorithm for generic sparse systems. For arbitrary sparse systems of n polynomials in n variables with fixed supports, we obtain an upper bound for the degree of the affine variety defined and we present an algorithm which computes finite sets of points representing its equidimensional components.