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. While generalizing the classical theory of Gr "o bner bases, it is not clear how modern algorithms for computing Gr "o bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug e re. 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 Gr "o b-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.
The computation of tropical varieties over @math with trivial valuation is available in the Gfan package by Anders Jensen (see @cite_6 ), by using standard Gröbner bases computations. Yet, for computation of tropical varieties over general fields, with non-trivial valuation, such techniques are not readily available. Then Chan and Maclagan have developed in @cite_13 a way to extend the theory of Gröbner bases to take into account the valuation and allow tropical computations. Their theory of tropical Gröbner bases is effective and allows, with a suitable division algorithm, a Buchberger algorithm. Following their work, a Matrix-F5 algorithm has been proposed in @cite_11 .
@cite_13 @cite_6 @cite_11
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Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Groebner bases, but we also give examples showing that the resulting Groebner bases can be substantially smaller than traditional Groebner bases. In the case K = Q with the p-adic valuation the algorithms have been implemented in a Macaulay 2 package. 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. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a p-adic fields. We design a strategy to compute such tropical Grobner bases by adapting the Matrix-F5 algorithm. Two variants of the Matrix-F5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster. Our study is performed both over any exact field with valuation and some inexact fields like Qp or Fq [[t]]. In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Grobner bases when the valuation is non-trivial. Numerical examples are provided.