Gröbner bases over polytopal affinoid algebras

Moulay A. Barkatou (1, 2) , Lucas Legrand (1, 2) , Tristan Vaccon (3, 1)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 XLIM - XLIM
3 XLIM - XLIM

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 Gröbner bases for polytopal affinoid algebras that extends both Caruso et al.’s theory of Gröbner bases on Tate algebras and Pauer et al.’s theory of Gröbner bases on Laurent polynomials. We provide effective algorithms to compute Gröbner bases for both ideals of Laurent polynomials and ideals in polytopal affinoid algebras. Experiments with a Sagemath implementation are provided.

Mots clés: Algebraic algorithms; Gröbner bases; Tate algebra; Laurent polynomials; Tropical analytic geometry; Algorithms; CCS CONCEPTS Computing methodologies → Algebraic algorithms Algorithms Gröbner bases Tate algebra Laurent polynomials Tropical analytic geometry; CCS CONCEPTS; Computing methodologies → Algebraic algorithms Algorithms

Domaines: Calcul formel [cs.SC]; Algèbre commutative [math.AC]; Théorie des nombres [math.NT]

Moulay A. Barkatou, Lucas Legrand, Tristan Vaccon. Gröbner bases over polytopal affinoid algebras. 2024. ⟨hal-04510822⟩