Signature-based algorithms for Gröbner bases over Tate algebras

Xavier Caruso; Tristan Vaccon; Thibaut Verron

doi:10.1145/3373207.3404035

Signature-based algorithms for Gröbner bases over Tate algebras

Xavier Caruso, Tristan Vaccon, Thibaut Verron

Institute for Algebra

Research output: Chapter in Book/Report/Conference proceeding › Conference proceedings › peer-review

Abstract

Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gröbner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gröbner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidence.

Original language	English
Title of host publication	Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
Editors	Angelos Mantzaflaris
Pages	70
Number of pages	8
ISBN (Electronic)	9781450371001
DOIs	https://doi.org/10.1145/3373207.3404035
Publication status	Published - Jul 2020

Fields of science

101 Mathematics
101001 Algebra
101005 Computer algebra
101013 Mathematical logic
102031 Theoretical computer science

JKU Focus areas

Digital Transformation

Access to Document

10.1145/3373207.3404035

Cite this

@inproceedings{accd407b40444185b1aa38e6001d7bc2,

title = "Signature-based algorithms for Gr{\"o}bner bases over Tate algebras",

abstract = "Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gr{\"o}bner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gr{\"o}bner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidence.",

author = "Xavier Caruso and Tristan Vaccon and Thibaut Verron",

year = "2020",

month = jul,

doi = "10.1145/3373207.3404035",

language = "English",

pages = "70",

editor = "Angelos Mantzaflaris",

booktitle = "Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation",

}

TY - GEN

T1 - Signature-based algorithms for Gröbner bases over Tate algebras

AU - Caruso, Xavier

AU - Vaccon, Tristan

AU - Verron, Thibaut

PY - 2020/7

Y1 - 2020/7

N2 - Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gröbner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gröbner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidence.

AB - Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gröbner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gröbner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidence.

UR - https://www.scopus.com/pages/publications/85090329646

U2 - 10.1145/3373207.3404035

DO - 10.1145/3373207.3404035

M3 - Conference proceedings

SP - 70

BT - Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

A2 - Mantzaflaris, Angelos

ER -

Signature-based algorithms for Gröbner bases over Tate algebras

Abstract

Fields of science

JKU Focus areas

Access to Document

Other files and links

Cite this