标题： 通过在交换形式幂级数上的拓扑重写计算完备局部等特征诺特环 显示英文标题

标题： Computing in complete local equicharacteristic Noetherian rings via topological rewriting on commutative formal power series

摘要： 在交换代数中，格罗布纳基理论使得可以在任何给定可计算域上的有限生成代数中进行计算。 然而，对于非有限生成的代数，必须采用其他方法。 例如，根据科恩结构定理，形式幂级数理想的规范基提供了类似的前景，但仅限于剩余域是可计算的完备局部等特征环。 使用重写理论的语言，可以将格罗布纳基表征为所诱导的重写系统的归并性。 迄今为止，通过纯粹的代数工具已经证明，对于具有广义归并性概念的规范基，类似的表征也成立。 随后，利用这一结果证明了两种广义归并性质在形式幂级数的上下文中实际上是等价的，尽管其中一种在一般情况下严格强于另一种。 在本文中，我们提出替代证明，利用拓扑重写理论的新工具来恢复规范基的表征以及广义归并性质之间的等价性。 目标是扩展格罗布纳基理论与经典代数重写理论之间的类比，以及规范基理论与拓扑重写理论之间的类比。 显示英文摘要

摘要： In commutative algebra, the theory of Gr\"obner bases enables one to compute in any finitely generated algebra over a given computable field. For non-finitely generated algebras however, other methods have to be pursued. For instance, it follows from the Cohen structure theorem that standard bases of formal power series ideals offer a similar prospect but for complete local equicharacteristic rings whose residue field is computable. Using the language of rewriting theory, one can characterise Gr\"obner bases in terms of confluence of the induced rewriting system. It has been shown, so far via purely algebraic tools, that an analogous characterisation holds for standard bases with a generalised notion of confluence. Subsequently, that result is utilised to prove that two generalised confluence properties, where one is actually in general strictly stronger than the other, are actually equivalent in the context of formal power series. In the present paper, we propose alternative proofs making use of tools purely from the new theory of topological rewriting to recover both the characterisation of standard bases and the equivalence between generalised confluence properties. The objective is to extend the analogy between Gr\"obner basis theory together with classical algebraic rewriting theory and standard basis theory with topological rewriting theory.