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On FGLM Algorithms with Tate Algebras

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-03133590
Contributor : Thibaut Verron Connect in order to contact the contributor
Submitted on : Monday, February 8, 2021 - 11:58:09 AM
Last modification on : Saturday, December 4, 2021 - 3:44:01 AM
Long-term archiving on: : Sunday, May 9, 2021 - 6:14:47 PM

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  • HAL Id : hal-03133590, version 1
  • ARXIV : 2102.05324

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Xavier Caruso, Tristan Vaccon, Thibaut Verron. On FGLM Algorithms with Tate Algebras. International Symposium on Symbolic and Algebraic Computation — ISSAC 2021, Jul 2021, Virtual event, Russia. ⟨hal-03133590⟩

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