Tate代数上Gr{<$}bner基的基于符号的算法
Signature-based algorithms for Gr{ö}bner bases over Tate algebras
摘要：
Tate在[Ta 71]中介绍了Tate代数，Tate代数在-adics上的解析几何中起着重要作用，在那里它们作为经典代数几何中多项式代数的对应物。在[CVV 19]中，Tate代数上的Gr{<$}bner基的形式被引入并有效地实现了.算法中的瓶颈之一是约简所花费的时间，这比多项式的代价要大得多。本文介绍了Tate代数的两种基于签名的Gr{<$}bner基算法，以避免大量的约简.它们已在SageMath中实现。我们讨论了他们的优势的基础上的数值证据。
英文摘要：
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{ö}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{ö}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.
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