Matrix-F5 algorithms and tropical Gröbner bases computation

Tristan Vaccon (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gröbner bases taking into account the valuation of $K$. We design a strategy to compute such tropical Gröbner bases by adapting the Matrix-F5 algorithm. We show that both Matrix-F5 and the signature-preserving Matrix-F5 are available to tropical computation with respective modifications. Our study is performed both over any exact field with valuation and some inexact fields like $\mathbb{Q}_p$ or $\mathbb{F}_q [[t]].$ In the latter case, we track the loss in precision, and show that the numerical stability compare favorably to the case of classical Gröbner bases. Numerical examples are provided.

Keywords: Gröbner bases; tropical geometry; F5 algorithm; $p$-adic precision; $p$-adic algorithm

Domains: Symbolic Computation [cs.SC]; Commutative Algebra [math.AC]

Tristan Vaccon. Matrix-F5 algorithms and tropical Gröbner bases computation. 2014. ⟨hal-00951953v1⟩