  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.
