On the computation of Gr\"obner bases for matrix-weighted homogeneous systems

TL;DR

This paper explores the structure of matrix-weighted homogeneous systems and introduces algorithms and optimizations for computing their Gr"obner bases, supported by experimental results and complexity considerations.

Contribution

It presents new linear algebra algorithms tailored for matrix-weighted homogeneous systems and discusses their complexity and optimization techniques.

Findings

Algorithms efficiently compute Gr"obner bases for weighted homogeneous systems.

Experimental results demonstrate the effectiveness of the proposed methods.

Discussion on regularity and its generic properties in these systems.

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 Gr\"obner bases. We present several linear algebra algorithms for computing Gr\"obner 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.