标题： 夹层伯恩斯坦-萨托多项式和伯恩斯坦不等式 显示英文标题

标题： Sandwich Bernstein-Sato Polynomials and Bernstein's Inequality

摘要： 伯恩斯坦不等式是光滑流形上$D$模块理论中的核心结果。 虽然伯恩斯坦不等式在一般奇点上的微分算子环上不成立，但Álvarez Montaner、Hernández、Jeffries、Núñez-Betancourt、Teixeira 和 Witt 的最新工作建立了在零特征下有限群不变量以及正特征中某些其他轻微奇点上的伯恩斯坦不等式。 受将此结果扩展到新的奇点环类的动机驱动，我们引入了一个称为“夹层伯恩斯坦-萨托多项式”的“双侧”类似物。 我们将这一概念应用于给出验证伯恩斯坦不等式的有效准则，并应用此准则来证明伯恩斯坦不等式对于$\mathbb{P}^a \times \mathbb{P}^b$的坐标环通过 Segre 嵌入成立。 我们还建立了一些关于夹层伯恩斯坦-萨托多项式的例子和基本结果。 显示英文摘要

摘要： Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of \`{A}lvarez Montaner, Hern\'andez, Jeffries, N\'u\~nez-Betancourt, Teixeira, and Witt establishes Bernstein's inequality for invariants of finite groups in characteristic zero and certain other mild singularities in positive characteristic. Motivated by extending this result to new classes of singular rings, we introduce a ``two-sided'' analogue of the Bernstein-Sato polynomial which we call the sandwich Bernstein-Sato polynomial. We apply this notion to give an effective criterion to verify Bernstein's inequality, and apply this to show that Bernstein's inequality holds for the coordinate ring of $\mathbb{P}^a \times \mathbb{P}^b$ via the Segre embedding. We also establish a number of examples and basic results on sandwich Bernstein-Sato polynomials.