On Affine Tropical F5 Algorithms
Tristan Vaccon (XLIM-MATHIS), Thibaut Verron, Kazuhiro Yokoyama

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 be a field equipped with a valuation. Tropical varieties over can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of .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 -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 -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 -adics.
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