An algorithm for annihilator and Bernstein-Sato polynomial of a rational function
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arXiv
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| Format: | Preprint |
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2026
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| author | González-Villa, Manuel León-Cardenal, Edwin Levandovskyy, Viktor Martín-Morales, Jorge |
| author_facet | González-Villa, Manuel León-Cardenal, Edwin Levandovskyy, Viktor Martín-Morales, Jorge |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An algorithm for annihilator and Bernstein-Sato polynomial of a rational function González-Villa, Manuel León-Cardenal, Edwin Levandovskyy, Viktor Martín-Morales, Jorge Algebraic Geometry Symbolic Computation 14F10, 16S32, 68W30, 32S40 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. |
| title | An algorithm for annihilator and Bernstein-Sato polynomial of a rational function |
| topic | Algebraic Geometry Symbolic Computation 14F10, 16S32, 68W30, 32S40 |
| url | https://arxiv.org/abs/2602.00280 |