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.

翻译：在本文中,我们检查了几个加权系统的加权同质系统的结构,以及它如何影响Gr\'obner基数的计算。我们提出了不同的方法来计算具有这一结构的系统的Gr\'obner基数,或者直接计算,或者缩小到现有结构。我们还提出了适合这一结构的优化技术。计算具有这一结构的系统的“Gr\'obner基数”的最自然顺序是按加权系统进行加权排序,我们讨论了使用算法直接计算这种顺序的基础的可能性,而不管系统的结构如何。我们讨论了可用于评估算法复杂性的常规概念,并证明这些概念是通用的,如果不是空的。最后,我们从SageMath的原型算法实施中提供了实验数据。