# Gr{\"o}bner bases over Tate algebras

## TL;DR

This paper extends the theory of Gr{"o}bner bases to Tate algebras, providing criteria, algorithms, and implementation details for computational purposes in p-adic analytic geometry.

## Contribution

It develops the formalism of Gr{"o}bner bases over Tate algebras, including criteria, algorithms, and implementation, bridging algebraic geometry and p-adic analysis.

## Key findings

- Proved an analogue of the Buchberger criterion for Tate algebras
- Designed Buchberger-like and F4-like algorithms for Tate algebras
- Discussed implementation in SageMath

## Abstract

Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{\"o}bner bases for Tate algebras. We prove an analogue of the Buchberger criterion in our framework and design a Buchberger-like and a F4-like algorithm for computing Gr{\"o}bner bases over Tate algebras. An implementation in SM is also discussed.