Bernstein-Sato theory modulo $p^m$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866911718833127424 |
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| author | Bitoun, Thomas Quinlan-Gallego, Eamon |
| author_facet | Bitoun, Thomas Quinlan-Gallego, Eamon |
| contents | For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z} / p^m$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$p$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $p$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$p$ reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernstein-Sato theory modulo $p^m$ Bitoun, Thomas Quinlan-Gallego, Eamon Commutative Algebra Algebraic Geometry Number Theory For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z} / p^m$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$p$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $p$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$p$ reduction. |
| title | Bernstein-Sato theory modulo $p^m$ |
| topic | Commutative Algebra Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2401.07082 |