On FGLM Algorithms with Tate Algebras

TL;DR

This paper extends the FGLM algorithm to Tate algebras, enabling efficient change of ordering and radii of convergence, thereby improving Gr{"o}bner basis computations in non-Archimedean analytic geometry.

Contribution

It introduces a novel FGLM algorithm adaptation for Tate algebras, facilitating faster computations and foundational tools for further algorithmic improvements.

Findings

Efficient change of ordering in Tate algebras achieved.

Algorithm for changing radii of convergence developed.

Potential acceleration of Gr{"o}bner basis computations demonstrated.

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 Gr{\"o}bner 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 Gr{\"o}bner 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.