publication . Conference object . Preprint . 2021

On FGLM Algorithms with Tate Algebras

Xavier Caruso; Tristan Vaccon; Thibaut Verron;
Open Access
  • Published: 18 Jul 2021
  • Publisher: ACM
  • Country: France
Abstract
International audience; Tate introduced in [Ta71] the notion of Tate algebras to serve, in the context of analytic geometry over the-adics, as a counterpart of polynomial algebras in classical algebraic geometry. In [CVV19, CVV20] the formalism of Gröbner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed. In the present article, we extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for fast change of ordering, this strategy has two other important benefits. First, it provides an efficient algorithm for changing the radii of convergence which, in particular, makes effective the bridge between the polynomial setting and the Tate setting and may help in speeding up the computation of Gröbner basis over Tate algebras. Second, it gives the foundations for designing a fast algorithm for interreduction, which could serve as basic primitive in our previous algorithms and accelerate them significantly.
Persistent Identifiers
Subjects
arXiv: Mathematics::Number Theory
ACM Computing Classification System: ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONGeneralLiterature_INTRODUCTORYANDSURVEY
free text keywords: p-adic precision, FGLM algorithm, Tate algebra, Gröbner bases, Algorithms, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Computer Science - Symbolic Computation, Context (language use), Formalism (philosophy), Signature (topology), Polynomial, Algorithm, Gröbner basis, Mathematics, Convergence (routing), Analytic geometry, Algebraic geometry
Funded by
ANR| CLap-CLap
Project
CLap-CLap
The p-adic Langlands correspondence: a constructive and algorithmic approach
  • Funder: French National Research Agency (ANR) (ANR)
  • Project Code: ANR-18-CE40-0026
,
FWF| Integral D-finite Functions
Project
  • Funder: Austrian Science Fund (FWF) (FWF)
  • Project Code: P 31571
  • Funding stream: Einzelprojekte
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