Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

TL;DR

This paper adapts a randomized geometric optimization algorithm to computational algebra, enabling efficient solutions for large polynomial systems and ideal generation by leveraging violator spaces and Helly-type theorems.

Contribution

It introduces the application of Clarkson's sampling algorithm to algebraic problems, establishing violator spaces for polynomial ideals and providing linear expected runtime algorithms.

Findings

Expected runtime is linear in the number of input polynomials.

Applicable to large-scale polynomial systems with small rank.

Utilizes Helly-type theorems for algebraic varieties.

Abstract

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).