Bernstein-Sato functional equations for ideals in positive characteristic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910993581342720 |
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| author | Tao, Siyong Xiao, Zida Zuo, Huaiqing |
| author_facet | Tao, Siyong Xiao, Zida Zuo, Huaiqing |
| contents | For an ideal of a regular $\cc$-algebra, its Bernstein-Sato polynomial is the monic polynomial of the lowest degree satisfying an Bernstein-Sato functional equation. We generalize the notion of Bernstein-Sato functional equations to the case of ideals in an $F$-finite ring of positive characteristic $p$, and show the relationship between these equations and Bernstein-Sato roots. By applying this theory, we provide an explicit description of Bernstein-Sato roots of a weighted homogeneous polynomial with an isolated singularity at the origin in characteristic $p$. Moreover, we give multiplicative and additive Thom-Sebastiani properties for the set of Bernstein-Sato roots, which prove the characteristic $p$ analogue of Budur and Popa's question. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20188 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernstein-Sato functional equations for ideals in positive characteristic Tao, Siyong Xiao, Zida Zuo, Huaiqing Commutative Algebra Algebraic Geometry For an ideal of a regular $\cc$-algebra, its Bernstein-Sato polynomial is the monic polynomial of the lowest degree satisfying an Bernstein-Sato functional equation. We generalize the notion of Bernstein-Sato functional equations to the case of ideals in an $F$-finite ring of positive characteristic $p$, and show the relationship between these equations and Bernstein-Sato roots. By applying this theory, we provide an explicit description of Bernstein-Sato roots of a weighted homogeneous polynomial with an isolated singularity at the origin in characteristic $p$. Moreover, we give multiplicative and additive Thom-Sebastiani properties for the set of Bernstein-Sato roots, which prove the characteristic $p$ analogue of Budur and Popa's question. |
| title | Bernstein-Sato functional equations for ideals in positive characteristic |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2410.20188 |