# Gröbner bases in Tate algebras

#### Séminaire AMAC: CASC

25/02/2021 - 09:30
Thibaut Verron *(Johannes Kelper Universität Linz)*

Tate series are a generalization of polynomials introduced by John Tate in 1962, when defining a p-adic analogue of the correspondence between algebraic geometry and analytic geometry. This p-adic analogue is called rigid geometry, and Tate series, similar to analytic functions in the complex case, are its fundamental objects. Tate series are defined as multivariate formal power series over a p-adic ring or field, with a convergence condition on a closed ball. Tate series are naturally approximated by multivariate polynomials over F_p or Z/p^n Z, and it is possible to define a theory of Gröbner bases for ideals of Tate series, which follows very closely that approximation. The implementation of such a theory then opens the way towards effective rigid geometry. In this talk, I will explain how to define and compute Gröbner bases for ideals of Tate series, with adapted versions of algorithms such as Buchberger's algorithm and the FGLM algorithm. (Joint work with Xavier Caruso and Tristan Vaccon) * Link for the video: https://meet-ljk.imag.fr/b/pie-sxg-kmz-6fm