标题： 奇异环的正特征伯恩斯坦-萨托理论 显示英文标题

标题： Bernstein-Sato theory for singular rings in positive characteristic

摘要： 伯恩斯坦-萨托多项式是特征零的多项式环或幂级数环中一个元素或理想的重要不变量，它与消失轨迹的奇点的各种代数和拓扑方面有有趣的联系。穆斯塔塔的工作后来由比通和第三作者扩展，为正特征的正规环提供了类似的伯恩斯坦-萨托理论。在本文中，我们将这一理论扩展到正特征的奇异环境环中。我们建立了大量奇异环的伯恩斯坦-萨托根的有限性和有理性结果，并将这些根与其他通过弗罗贝尼乌斯映射定义的数值不变量相关联。我们还在正规情况下获得了若干新的结果和简化的论证。 显示英文摘要

摘要： The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Musta\c{t}\u{a}, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case.