On FGLM Algorithms with Tropical Gröbner bases

09/04/2020
by   Yuki Ishihara, et al.
0

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. 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 Gröbner 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 Gröbner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gröbner basis to a classical Gröbner basis. We provide tools to chain the stable computation of a tropical Gröbner 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/16/2018

On Affine Tropical F5 Algorithms

Let K be a field equipped with a valuation. Tropical varieties over K ca...
research
02/02/2016

On the p-adic stability of the FGLM algorithm

Nowadays, many strategies to solve polynomial systems use the computatio...
research
05/16/2017

A Tropical F5 algorithm

Let K be a field equipped with a valuation. Tropical varieties over K ca...
research
05/09/2017

Improved Computation of Involutive Bases

In this paper, we describe improved algorithms to compute Janet and Pomm...
research
02/10/2021

On FGLM Algorithms with Tate Algebras

Tate introduced in [Ta71] the notion of Tate algebras to serve, in the c...
research
11/10/2015

A unifying form for noetherian polynomial reductions

Polynomial reduction is one of the main tools in computational algebra w...
research
01/10/2021

A New Type of Bases for Zero-dimensional Ideals

We formulate a substantial improvement on Buchberger's algorithm for Grö...

Please sign up or login with your details

Forgot password? Click here to reset