Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Short proofs of ideal membership

Clemens Hofstadler, Thibaut Verron

TL;DR

The paper tackles sparsest cofactor representations for noncommutative polynomial ideals, framing ideal membership as certificates and proving that bounded-term representations are decidable yet NP-complete. It introduces a practical pipeline that leverages signature-based Gröbner bases to constrain the search space and reduces the problem to a linear program that minimizes either the $\ell_0$ or $\ell_1$ norm, with theoretical guarantees under certain ideal classes. The approach yields sparser representations than traditional Gröbner-based proofs and demonstrates substantial practical gains via a SageMath prototype, including a fully polynomial-time LP step and powerful pruning of the syzygy space. This work links noncommutative ideal membership, syzygy computation, and sparse linear optimization, offering scalable methods for producing short, verifiable proofs of operator statements translated into polynomial identities.

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 Gröbner 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.

Short proofs of ideal membership

TL;DR

The paper tackles sparsest cofactor representations for noncommutative polynomial ideals, framing ideal membership as certificates and proving that bounded-term representations are decidable yet NP-complete. It introduces a practical pipeline that leverages signature-based Gröbner bases to constrain the search space and reduces the problem to a linear program that minimizes either the or norm, with theoretical guarantees under certain ideal classes. The approach yields sparser representations than traditional Gröbner-based proofs and demonstrates substantial practical gains via a SageMath prototype, including a fully polynomial-time LP step and powerful pruning of the syzygy space. This work links noncommutative ideal membership, syzygy computation, and sparse linear optimization, offering scalable methods for producing short, verifiable proofs of operator statements translated into polynomial identities.

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 Gröbner 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.
Paper Structure (13 sections, 17 theorems, 14 equations, 1 table, 3 algorithms)

This paper contains 13 sections, 17 theorems, 14 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free algebra
  4. Free bimodule
  5. Signature-based algorithms
  6. Decidability and complexity
  7. Computing sparse representations
  8. Finding a suitable basis $B$
  9. Detecting redundant syzygies
  10. Translation into linear program
  11. Putting everything together
  12. Special case: totally unimodular matrices
  13. Experimental results

Key Result

Theorem 2.6

There exists an algorithm to correctly enumerate a Gröbner basis of $\textnormal{Syz}(I^{[\Sigma]})$ in increasing signature order. In particular, for all $\sigma \in M(\Sigma)$, stopping the algorithm at the first syzygy with signature $\succeq \sigma$ yields a finite Gröbner basis of $\textnormal{

Theorems & Definitions (49)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Remark 3.1
  • Remark 3.2
  • ...and 39 more