Séminaire Calcul Formel

On Polynomial Ideals and Overconvergence in Tate Algebras

by Dr Tristan Vaccon (Université de Limoges)

XR203 (XLIM)



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. The collection of such series forms a Tate algebra.
Polynomials are Tate series for any such convergence condition.
We show that for an ideal in a Tate algebra generated by polynomials, we can compute a Gröbner basis made of polynomials.
This generalize to the case of an ideal in a Tate algebra generated by overconverging Tate series, i.e. Tate series converging on bigger balls.
In addition, we prove the existence of an analytic universal Gröbner basis for a polynomial ideal : a finite set of polynomial such that it is a Gröbner basis for the completion of this ideal in any Tate algebra.

This is a joint work with Xavier Caruso and Thibaut Verron.