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Gr{ö}bner bases over polytopal affinoid algebras

Moulay A. Barkatou, Lucas Legrand, Tristan Vaccon

TL;DR

The article develops a Gröbner bases theory for polytopal affinoid algebras $K\{\mathbf{X};P\}$, bridging rigid analytic geometry and tropical analytic geometry. It extends GB theory from Tate algebras and Laurent polynomials to polytopal settings by employing generalized monomial orders built from conic decompositions and by adapting division, S-pairs, and Buchberger's algorithm to both $K\{\mathbf{X};r\}$ and $K\{\mathbf{X};P\}$. The work provides effective division and Gröbner basis algorithms, including procedures to compute the necessary leading monomials via polyhedral computations and conic decompositions, with a SageMath implementation and demonstrations. This framework enables computation of GBs in Laurent-domain and polytope-constrained algebras, supporting practical tropical-analytic applications in non-archimedean geometry.

Abstract

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

Gr{ö}bner bases over polytopal affinoid algebras

TL;DR

The article develops a Gröbner bases theory for polytopal affinoid algebras , bridging rigid analytic geometry and tropical analytic geometry. It extends GB theory from Tate algebras and Laurent polynomials to polytopal settings by employing generalized monomial orders built from conic decompositions and by adapting division, S-pairs, and Buchberger's algorithm to both and . The work provides effective division and Gröbner basis algorithms, including procedures to compute the necessary leading monomials via polyhedral computations and conic decompositions, with a SageMath implementation and demonstrations. This framework enables computation of GBs in Laurent-domain and polytope-constrained algebras, supporting practical tropical-analytic applications in non-archimedean geometry.

Abstract

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{ö}bner bases for polytopal affinoid algebras that extends both Caruso et al.'s theory of Gr{ö}bner bases on Tate algebras and Pauer et al.'s theory of Gr{ö}bner bases on Laurent polynomials. We provide effective algorithms to compute Gr{ö}bner bases for both ideals of Laurent polynomials and ideals in polytopal affinoid algebras. Experiments with a Sagemath implementation are provided.
Paper Structure (23 sections, 25 theorems, 48 equations, 5 figures, 5 algorithms)

This paper contains 23 sections, 25 theorems, 48 equations, 5 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Setting
  3. Generalized monomial order
  4. Multivariate division in $K\{\mathbf{X};r\}$
  5. The monoid of terms $T\{\mathbf{X}\}$
  6. Multivariate division algorithm
  7. Gröbner bases for ideals in $K\{\mathbf{X};r\}$
  8. Gröbner bases
  9. S-pairs and Buchberger criterion
  10. Buchberger's algorithm
  11. Implementation
  12. Computation of the $T_i(f)$'s
  13. A particular conic decomposition
  14. Gröbner bases for ideals in $K\{\mathbf{X};P\}$
  15. New definitions
  16. ...and 8 more sections

Key Result

Lemma 3.6

Let $(T_i)_{i \in I}$ be a conic decomposition of $T$ and $E$ be either $\{1\}$ or one of the $T_i$. Let $<_G$ be a total group order on $T$ (e.g. the lexicographical order). Let $\phi : T \to \mathbb{Q}_{\ge 0}$ be a function fulfilling the following conditions: Then the order $<$ defined by is a g.m.o on $T$ for the decomposition $(T_i)_{i \in I}$.

Figures (5)

  • Figure 1: Conic decompositions for $n=2$
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:

Theorems & Definitions (86)

  • Definition 3.1: PU:1999
  • Example 3.2
  • Example 3.3
  • Definition 3.4: PU:1999
  • Remark 3.5
  • Lemma 3.6: PU:1999
  • Definition 3.7
  • Example 3.8
  • Example 3.9
  • Example 3.10
  • ...and 76 more