Polyhedral affinoid algebras have been introduced by Einsiedler, Kapranov and Lind to connect rigid analytic geometry (analytic geometry over non-archimedean fields) and tropical geometry.In this article, we present a theory of Gr{\"o}bner bases for polytopal affinoid algebras that extends both Caruso et al.'s theory of Gr{\"o}bner bases on Tate algebras and Pauer et al.'s theory of Gr{\"o}bner bases on Laurent polynomials.We provide effective algorithms to compute Gr{\"o}bner bases for both ideals of Laurent polynomials and ideals in polytopal affinoid algebras. Experiments with a Sagemath implementation are provided.

翻译：Einsiedler、Kapranov和Lind引入了多胞仿射代数，以连接刚性解析几何（非阿基米德域上的解析几何）与热带几何。本文提出了一种适用于多胞仿射代数的Gröbner基理论，该理论既推广了Caruso等人关于Tate代数的Gröbner基理论，也推广了Pauer等人关于Laurent多项式的Gröbner基理论。我们提供了有效算法，用于计算Laurent多项式理想和多胞仿射代数中的Gröbner基。文中还给出了基于Sagemath实现的实验结果。