3 research outputs found
A Tropical F5 algorithm
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
Gröbner bases over fields with valuations
Let K be a field with a valuation and let S be the polynomial ring S := K[x1; : : : ; xn]. We discuss the extension of Gröbner 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 Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case K = Q with the p-adic valuation the algorithms have been implemented in a Macaulay 2 package