An algorithm for annihilator and Bernstein-Sato polynomial of a rational function

TL;DR

This paper introduces an algorithm to compute the Bernstein-Sato polynomial of rational functions, enabling the analysis of singularities and providing explicit examples to support conjectures.

Contribution

The authors develop a novel algorithm for calculating the Bernstein-Sato polynomial of rational functions using annihilator computation and implement it in SINGULAR.

Findings

Algorithm successfully computes non-trivial Bernstein-Sato polynomials.

Implementation in SINGULAR facilitates practical computations.

Supports existing conjectures in singularity theory.

Abstract

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 Gr\"obner bases in noncommutative PBW algebras. The algorithm allows us to exhibit some explicit non-trivial examples and to support some existing conjectures.