Signature-based algorithms for Gröbner bases over tate algebras

Abstract

Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the p-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gröbner bases over Tate algebras has been introduced and effectively implemented. One of the bottlenecks in the algorithms was the time spent on reduction, which are significantly costlier than over polynomials. In the present article, we introduce two signature-based Gröbner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidence.