Sparse systems and algorithmic equidimensional decomposition

TL;DR

This paper introduces a probabilistic algorithm for identifying and analyzing equidimensional components of algebraic varieties defined by sparse polynomial systems, with complexity depending on the supports' combinatorial properties.

Contribution

It presents a novel probabilistic method to compute witness sets for equidimensional components of sparse polynomial systems, with polynomial complexity.

Findings

Algorithm effectively characterizes equidimensional components.

Complexity is polynomial in combinatorial invariants.

Provides a new tool for algebraic geometry analysis.

Abstract

We present a new probabilistic algorithm that characterizes the equidimensional components of the affine algebraic variety defined by an arbitrary sparse polynomial system with prescribed supports. For each equidimensional component, the algorithm computes a witness set, namely a finite set obtained by intersecting the component with a generic linear variety of complementary dimension. The complexity of the algorithm is polynomial in combinatorial invariants associated to the supports of the polynomials involved.