The singularity theory of rational functions, i.e., the quotient of two polynomials, has been investigated in the past two decades. The Bernstein-Sato polynomial of a rational function has recently been introduced by Takeuchi. However, only trivial examples are known. We provide an algorithm for computing the Bernstein-Sato polynomial in this context. The strategy is to compute the annihilator of the rational function by using the annihilator of the pair consisting of the numerator and denominator of the quotient. In a natural way a non-vanishing condition on the Bernstein-Sato ideal of the pair appears. This method has been implemented in freely available computer algebra system SINGULAR. It relies on Gröbner bases in noncommutative PBW algebras. The algorithm allows us to exhibit some explicit non-trivial examples and to support some existing conjectures.

翻译：有理函数（即两个多项式之商）的奇点理论在过去二十年中得到了广泛研究。Takeuchi 最近引入了有理函数的 Bernstein-Sato 多项式，但迄今仅知一些平凡示例。本文提出在此背景下计算 Bernstein-Sato 多项式的一种算法。其核心策略是通过利用商函数分子与分母构成的数对之消元子，来计算该有理函数的消元子。在此过程中，数对的 Bernstein-Sato 理想需满足非零条件的要求会自然显现。该方法已在开源计算机代数系统 SINGULAR 中实现，其计算基础为非交换 PBW 代数中的 Gröbner 基。该算法不仅使我们能展示若干显式的非平凡示例，还为一些现有猜想提供了佐证。