On the computation of Gröbner bases for pluriweighted-homogeneous systems

In this paper, we examine the structure of systems which are weighted homogeneous for several systems of weights, and how it impacts Gr\"obner basis computations. We present different ways to compute Gr\"obner bases for systems with this structure, either directly or by reducing to existing structures. We also present optimization techniques which are suitable for this structure. The most natural orderings to compute a Gr\"obner basis for systems with this structure are weighted orderings following the systems of weights, and we discuss the possibility to use the algorithms in order to directly compute a basis for such an order, regardless of the structure of the system. We discuss applicable notions of regularity which could be used to evaluate the complexity of the algorithm, and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.