Short proofs of ideal membership

TL;DR

This paper investigates the problem of finding the sparsest cofactor representations of polynomial ideal elements, providing complexity results and a practical linear optimization approach that often yields near-optimal solutions.

Contribution

It introduces a decidable and NP-complete problem for sparse cofactor representations, and proposes a practical algorithm leveraging signature-based Gr"obner basis techniques.

Findings

The problem of bounded-term cofactor representations is NP-complete.

The proposed algorithm produces sparser representations in practice.

For certain ideals, the method achieves optimal representations.

Abstract

A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting. We show that the problem of computing cofactor representations with a bounded number of terms is decidable and NP-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gr\"obner basis algorithms. We show that for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.