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On the computation of Gröbner bases for matrix-weighted homogeneous systems

Thibaut Verron

TL;DR

The paper addresses computing Gröbner bases for systems that are matrix-weighted homogeneous, a structure generalizing multihomogeneous and weighted homogeneous cases. It develops a general Matrix-F5 framework tailored to matrix-weighted gradings, with specialized variants that reduce to multihomogeneous and sparse settings and includes optimization strategies such as parallelization and gcd-based pruning. It analyzes regularity notions via Hilbert multiseries, proves conditional genericity results, and demonstrates practical gains through a SageMath prototype. The work advances structured polynomial solving by exploiting multidimensional weight structures to improve efficiency and scale for complex symbolic computations with potential applications in physics and beyond.

Abstract

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques. As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.

On the computation of Gröbner bases for matrix-weighted homogeneous systems

TL;DR

The paper addresses computing Gröbner bases for systems that are matrix-weighted homogeneous, a structure generalizing multihomogeneous and weighted homogeneous cases. It develops a general Matrix-F5 framework tailored to matrix-weighted gradings, with specialized variants that reduce to multihomogeneous and sparse settings and includes optimization strategies such as parallelization and gcd-based pruning. It analyzes regularity notions via Hilbert multiseries, proves conditional genericity results, and demonstrates practical gains through a SageMath prototype. The work advances structured polynomial solving by exploiting multidimensional weight structures to improve efficiency and scale for complex symbolic computations with potential applications in physics and beyond.

Abstract

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques. As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.
Paper Structure (21 sections, 16 theorems, 30 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 21 sections, 16 theorems, 30 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgements
  4. Funding
  5. Notations
  6. Definitions
  7. Matrix gradings
  8. Size-bounded matrix of weights
  9. Matrix-weighted degree-reverse-lexicographical ordering
  10. Matrix-F5 algorithm
  11. General Matrix-F5 algorithm for structured systems
  12. Matrix-weighted homogeneous Matrix-F5 algorithm
  13. Optimizations
  14. Reducing to multihomogeneous systems
  15. Sparse Gröbner basis algorithms
  16. ...and 6 more sections

Key Result

Proposition 3.9

Let $\mathbf{W}_{1}, \mathbf{W}_{2}$ be two matrices of weights. Then the following are equivalent:

Figures (1)

  • Figure 1: Coefficients of the series $(1-T^{4}U^{6})\mathsf{HS}_{A/I_{2}}$ (in black and gray) and $\mathsf{HS}_{A/I_{3}}(T,U)$ (in black only), in Example \ref{['ex:semi-reg-hs']}

Theorems & Definitions (60)

  • Definition 3.1
  • Definition 3.2: Kreuzer-2005-ComputationalCommutativeAlgebra
  • Remark 3.3
  • Example 3.4
  • Definition 3.5
  • Example 3.6
  • Remark 3.7
  • Definition 3.8
  • Proposition 3.9
  • proof
  • ...and 50 more