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PreprintCC BY-NC-ND 4.0arXiv

Bernstein-Sato polynomials of generic Pfaffian ideals

  • Yifan Chen Huaiqing Zuo
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Abstract

We determine all roots and their multiplicities in the Bernstein--Sato polynomial of every generic Pfaffian ideal. In the nonmaximal case, the roots lie in explicit rational windows indexed by the rank strata. Integral roots retain their label multiplicities, whereas every nonintegral root is simple, including values with several labels. The proof combines graph-module relations, skew Capelli operators, central characters, and a weighted Mellin integral. We deduce the strong motivic monodromy conjecture for Pfaffian ideals, with pole orders bounded by Bernstein--Sato root multiplicities. For Pfaffians of size 2r, the largest possible root multiplicity as the matrix size varies is ⌊(√(8r+1)−1)/2⌋. In the nonmaximal case we also prove that the nonunipotent part of the Verdier specialization has zero nilpotent monodromy. Keywords. Bernstein-Sato polynomial, Pfaffian ideal, root multiplicities, strong monodromy conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{zuo2026bernsteinsato,
  title = {{Bernstein-Sato polynomials of generic Pfaffian ideals}},
  author = {Yifan Chen Huaiqing Zuo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07571},
  eprint = {2611.07571},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07571},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Bernstein-Sato polynomials of generic Pfaffian ideals
AU  - Yifan Chen Huaiqing Zuo
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07571
UR  - https://arxiv.org/abs/2611.07571
AB  - We determine all roots and their multiplicities in the Bernstein--Sato polynomial of every generic Pfaffian ideal. In the nonmaximal case, the roots lie in explicit rational windows indexed by the rank strata. Integral roots retain their label multiplicities, whereas every nonintegral root is simple, including values with several labels. The proof combines graph-module relations, skew Capelli operators, central characters, and a weighted Mellin integral. We deduce the strong motivic monodromy conjecture for Pfaffian ideals, with pole orders bounded by Bernstein--Sato root multiplicities. For Pfaffians of size 2r, the largest possible root multiplicity as the matrix size varies is ⌊(√(8r+1)−1)/2⌋. In the nonmaximal case we also prove that the nonunipotent part of the Verdier specialization has zero nilpotent monodromy. Keywords. Bernstein-Sato polynomial, Pfaffian ideal, root multiplicities, strong monodromy conjecture.
ER  -