$b$-函数有理根的检验算法及其应用
Algorithms for Checking Rational Roots of $b$-functions and their Applications
摘要：
超曲面的Bernstein-Sato多项式是一个重要的研究对象，有着广泛的应用。众所周知，通过计算获得它是复杂的，因为许多开放的问题和挑战表明。本文提出了一个算法族\texttt{checkRoot}，用于优化检验给定的有理数是否是Bernstein-Sato多项式的根及其重数的计算。该算法用于计算完整理想关于权的整体或局部Bernstein-Sato多项式和$b$-函数。它们被应用在许多情况下，其中有可能计算多项式的上界。也就是说，它可以通过嵌入分辨率，拓扑等价的奇点或使用A'Campo公式和谱数来实现。我们还提出了对数比较问题和相交同调D-模的方法。几个应用程序以及解决方案的一些挑战，这是棘手的经典方法。其中一个主要的应用包括计算的仿射空间的分层与当地的$b$-功能是恒定的每一层。值得注意的是，我们提出的算法不采用准素分解。我们也将我们的结果应用于计算Bernstein-Sato多项式的品种。本文中的方法已在{\sc Singular：Plural}中作为库{\tt dmod. lib}和{\tt bfun.lib}实现。本文中的所有例子都是用这种方法计算的。
英文摘要：
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called \texttt{checkRoot} for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and $b$-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A'Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications consists of computing of a stratification of affine space with the local $b$-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Also we apply our results for the computation of Bernstein-Sato polynomials for varieties. The methods from this paper have been implemented in {\sc Singular:Plural} as libraries {\tt dmod.lib} and {\tt bfun.lib}. All the examples from the paper have been computed with this implementation.
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