On Polynomial Ideals And Overconvergence In Tate Algebras

TL;DR

This paper investigates polynomial and overconvergent series ideals in Tate algebras, introducing algorithms for computing specialized Gröbner bases with size growth controlled by precision, and establishing the existence of universal bases.

Contribution

It proves polynomial ideals admit polynomial Gröbner bases, proposes algorithms for overconvergent bases, and demonstrates a universal Gröbner basis compatible with all convergence radii.

Findings

Polynomial ideals have polynomial Gröbner bases.

Algorithms for overconvergent bases with linear size growth.

Existence of a universal analytic Gröbner basis.

Abstract

In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{\"o}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm grows linearly with the precision. Following the same ideas, we propose an algorithm which computes an overconvergent basis for an ideal spanned by overconvergent series. Finally, we prove the existence of a universal analytic Gr{\"o}bner basis for polynomial ideals in Tate algebras, compatible with all convergence radii.