On FGLM Algorithms with Tropical Gr\"obner bases

TL;DR

This paper adapts the FGLM algorithm to the tropical setting, enabling efficient computation of tropical and classical Gröbner bases over valued fields, with improved stability for p-adic computations.

Contribution

It introduces new linear algebra algorithms for tropical FGLM, bridging tropical and classical Gröbner basis computations, and implements these methods in SageMath.

Findings

Tropical FGLM algorithms efficiently change weights for tropical Gröbner bases.

New algorithms improve stability in p-adic numerical computations.

Implementation demonstrates time-complexity and stability advantages.

Abstract

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. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm's incremental way, monomo-mial by monomial, to compute the multiplication matrices and the change of basis matrix can not be transposed at all to the tropical setting. We mitigate this issue by developing new linear algebra algorithms and apply them to our new tropical FGLM algorithms. Motivations are twofold. Firstly, to compute tropical varieties, one usually goes through the computation of many tropical Gr{\"o}bner bases defined for varying weights (and then varying term orders). For an ideal of dimension 0, the tropical FGLM algorithm provides an efficient way to go from a tropical Gr{\"o}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{\"o}bner basis to a classical Gr{\"o}bner basis. We provide tools to chain the stable computation of a tropical Gr{\"o}bner basis (for weight [0,. .. , 0]) with the p-adic stabilized variants of FGLM of [RV16] to compute a lexicographical or shape position basis. All our algorithms have been implemented into SageMath. We provide numerical examples to illustrate time-complexity. We then illustrate the superiority of our strategy regarding to the stability of p-adic numerical computations.