Gr{ö}bner bases over polytopal affinoid algebras

2024-03-20

International Symposium on Symbolic and Algebraic Computation

AI Summary

This work develops a Gröbner basis theory for polyhedral affinoid algebras, unifying the classical frameworks of Tate algebras and Laurent polynomial rings while bridging rigid analytic geometry and tropical geometry. To this end, we introduce—firstly—a weighted monomial order compatible with non-Archimedean valuations and polyhedral norms, along with a corresponding reduction mechanism; design a Gröbner basis algorithm based on this order; implement it in SageMath; and integrate tropical and rigid analytic structures via norm modeling techniques. Our main contributions are: (i) the first comprehensive Gröbner theory for polyhedral rigid analytic algebras; (ii) a computationally effective and verifiable Gröbner basis algorithm for ideals in both Laurent rings and polyhedral affinoid algebras; and (iii) open-source implementation—including full code and numerical benchmarks—to facilitate future research at the intersection of tropical and rigid analytic geometry.

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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{""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.

Problem

• Extend Gröbner bases theory to polytopal affinoid algebras
• Bridge rigid analytic and tropical geometry via algebra
• Develop algorithms for Gröbner bases in these algebras

Innovation

• Extends Groebner bases to polytopal affinoid algebras
• Connects rigid and tropical geometry via algebras
• Provides effective algorithms for Groebner computations