Abstract

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Grobner bases taking into account the valuation of K. While generalizing the classical theory of Grobner bases, it is not clear how modern algorithms for computing Grobner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugere.In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness.Because of the use of the valuation, the theory of tropical Grobner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.

Highlights

  • The theory of tropical geometry is only a few decades old

  • We show that following this definition, an F5 algorithms can be performed to compute tropical Gröbner bases

  • Our second objective in designing a tropical F5 algorithm is to pave the way for an algorithm that could at the same time be comparable to the fast methods for classical Gröbner bases, have a termination criterion and still benefit from the stability that can be obtained for tropical Gröbner bases

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Summary

A Tropical F5 algorithm

To cite this version: Tristan Vaccon, Kazuhiro Yokoyama. ISSAC’17-Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation, 2017, ￿10.1145/3087604.3087630￿. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

Introduction
Related works on tropical Gröbner bases
Main idea and results
Notations
Signature
Tropical S-Gröbner bases
Linear algebra and tropical S-Gröbner bases
F5 criterion
A tropical F5 algorithm
Implementation and numerical results
Future works
Full Text
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