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Article Dans Une Revue ISSAC '18: 2018 ACM International Symposium on Symbolic and Algebraic Computation Année : 2018

On Affine Tropical F5 Algorithms

Résumé

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 $K$. Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a $p$-adic field that are more stable than that of classical Gröbner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gröbner bases in an affine setting. We provide numerical examples to illustrate time-complexity and $p$-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 $p$-adics.
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Dates et versions

hal-01792165 , version 1 (15-05-2018)

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Tristan Vaccon, Thibaut Verron, Kazuhiro Yokoyama. On Affine Tropical F5 Algorithms. ISSAC '18: 2018 ACM International Symposium on Symbolic and Algebraic Computation, 2018, ⟨10.1145/3208976.3209012⟩. ⟨hal-01792165⟩
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