An algorithm for annihilator and Bernstein-Sato polynomial of a rational function
Manuel González-Villa, Edwin León-Cardenal, Viktor Levandovskyy, Jorge Martín-Morales
TL;DR
This work addresses computing the Bernstein-Sato polynomial of a rational function $f/g$ by reducing to the annihilator of $(f/g)^s$ and exploiting the annihilator of the pair $(f,g)$. It introduces a central-saturation framework and a sufficient $ ext{C}_m$-condition on the Bernstein-Sato ideal $B_{f,g}$ to recover the annihilator as a saturation of $I(s,-s-m)$, together with a syzygy-based remedy when the condition fails. An algorithm is given to compute $b^{(N)}_{f/g,m}(s)$ for fixed $N$, implemented in the computer algebra system SINGULAR using noncommutative Gröbner bases, enabling explicit nontrivial examples and connections to monodromy. The results illuminate links between singularity invariants, monodromy eigenvalues, and Bernstein-Sato data of rational functions, and provide evidence in support of existing conjectures.
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 Gröbner bases in noncommutative PBW algebras. The algorithm allows us to exhibit some explicit non-trivial examples and to support some existing conjectures.