On Gröbner bases over Tate algebras

Abstract: 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 opens the way towards effective rigid geometry.

In this talk, I will present classical algorithms to compute Gröbner bases (Buchberger, F5, FGLM) and how they can be adapted for Tate series.

Joint work with Xavier Caruso and Thibaut Verron.

algebraic geometrynumber theory

Audience: researchers in the topic

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SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.

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