Signature-based algorithms for Gröbner 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 Gröbner bases over Tate algebras has been introduced and e ectively implemented. One of the bottleneck in the algorithms was the time spent on reduction, which are signi cantly 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 S M. We discuss their superiority based on numerical evidences.

On FGLM Algorithms with Tate Algebras

“…In earlier papers [6,7], the authors showed that it is possible to define and compute Gröbner bases of Tate ideals with coefficients in Z or Q , and that the definitions are compatible with the usual theory on polynomials over the residue field F or over the coefficient ring. A major limitation of the algorithms designed in loc.…”

“…is the increasing cost of reductions as the precision grows. Our previous paper [7] addresses the case of expensive reductions to zero, through the use of signature algorithms, but computing the result of nontrivial reductions remains expensive. Another question left open was whether it is possible to exploit overconvergence properties, namely the knowledge that the series we are working with satisfy a stronger convergence condition.…”

“…

Tate introduced in [18] 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 [6,7] 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 [8] to Tate algebras.

…”

On FGLM Algorithms with Tate Algebras

“…In earlier papers [CVV19,CVV20], the authors showed that it is possible to define and compute Gröbner bases of Tate ideals with coefficients in Z or Q , and that the definitions are compatible with the usual theory on polynomials over the residue field F or over the coefficient ring. A major limitation of the algorithms is the increasing cost of reductions as the precision grows.…”

“…A major limitation of the algorithms is the increasing cost of reductions as the precision grows. Our previous paper [CVV20] addresses the case of expensive reductions to zero, through the use of signature algorithms, but computing the result of non-trivial reductions remains expensive. Another question left open was whether it is possible to exploit overconvergence properties, namely the knowledge that the series we are working with satisfy a stronger convergence condition.…”