Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gr{\"o}bner bases over Tate algebras has been introduced and effectively implemented. One of the bottleneck 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 Gr{\"o}bner bases algorithms for Tate algebras, in order to avoid many reductions. They have been implemented in SageMath. We discuss their superiority based on numerical evidences.

中文翻译：

---

泰特代数上 Gr{\"o}bner 基的基于签名的算法

Tate 在 [Ta71] 中介绍，Tate 代数在解析几何的上下文中比 the-adics 发挥重要作用，在那里它们充当经典代数几何中多项式代数的使用的对应物。在 [CVV19] 中，Gr{\"o}bner bases over Tate algebras 的形式主义已经被引入并有效实现。算法中的瓶颈之一是花费在减少上的时间，这比多项式花费的时间要多得多。在目前在这篇文章中，我们为 Tate 代数介绍了两种基于签名的 Gr{\"o}bner 基算法，以避免许多减少。它们已在 SageMath 中实现。我们根据数值证据讨论它们的优越性。