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

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.