On Affine Tropical F5 Algorithms

TL;DR

This paper extends the F5 algorithm to affine tropical Gr{"o}bner bases over valued fields, improving computational stability and efficiency for polynomial systems over p-adic fields, and demonstrating its potential as a precursor to FGLM algorithms.

Contribution

It introduces an affine tropical F5 algorithm, expanding tropical Gr{"o}bner basis computations beyond homogeneous polynomials, with practical stability and complexity benefits.

Findings

Demonstrates improved p-adic stability in computations

Provides numerical evidence of time-complexity benefits

Shows potential as a preliminary step for classical lex basis computation

Abstract

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of $K$.Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a $p$-adic field that are more stable than that of classical Gr{\"o}bner bases.Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gr{\"o}bner bases in an affine setting.We provide numerical examples to illustrate time-complexity and $p$-adic stability of this tropical F5 algorithm.We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over $p$-adics.