A Tropical F5 algorithm

05/16/2017

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by   Tristan Vaccon, et al.

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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. While generalizing the classical theory of Gröbner bases, it is not clear how modern algorithms for computing Gröbner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugère. 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 Gröb-ner 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.