The Multivariable Strong Monodromy Conjecture for Plane Curves

Abstract

We prove the multivariable Strong Monodromy Conjecture for the local topological zeta function of arbitrary tuples of nonzero nonunit plane curve germs. The entries may be nonreduced and may have common irreducible factors. We also prove that the non-equivariant local motivic and topological zeta functions have the same polar locus and generic pole orders, using a multivariable extension of the Nicaise--Xu definition of motivic pole order. Our arguments combine twisted Poincaré residues with resolution formulas and specialization to one variable. In arbitrary dimension, we prove that a nonzero iterated residue class on a stratum of a log resolution implies the vanishing of the Bernstein--Sato ideal at the corresponding parameter. In particular, every topological polar hyperplane whose order equals the ambient dimension is contained in this zero locus. For algebraic germs, the topological pole order is at most the motivic pole order, and the two attain maximal order simultaneously.

The strong monodromy conjecture for hyperplane arrangements
Lei Wu
arXiv6 Oct 2026PreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv
Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements
Dougal Davis, Ruijie Yang
arXiv19 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Analysis & PDEsarXiv
On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
Xiping Zhang
arXiv21 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves
Ryan Contreras, Joseph Cummings, Eric Riedl +1
arXiv23 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
A p-adic monodromy theorem for curves
Hansheng Diao, Yong Suk Moon, Zijian Yao
arXiv7 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{tan2026multivariable,
  title = {{The Multivariable Strong Monodromy Conjecture for Plane Curves}},
  author = {Sheng Tan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.26087},
  eprint = {2608.26087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26087},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The Multivariable Strong Monodromy Conjecture for Plane Curves
AU  - Sheng Tan
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.26087
UR  - https://arxiv.org/abs/2608.26087
AB  - We prove the multivariable Strong Monodromy Conjecture for the local topological zeta function of arbitrary tuples of nonzero nonunit plane curve germs. The entries may be nonreduced and may have common irreducible factors. We also prove that the non-equivariant local motivic and topological zeta functions have the same polar locus and generic pole orders, using a multivariable extension of the Nicaise--Xu definition of motivic pole order. Our arguments combine twisted Poincaré residues with resolution formulas and specialization to one variable. In arbitrary dimension, we prove that a nonzero iterated residue class on a stratum of a log resolution implies the vanishing of the Bernstein--Sato ideal at the corresponding parameter. In particular, every topological polar hyperplane whose order equals the ambient dimension is contained in this zero locus. For algebraic germs, the topological pole order is at most the motivic pole order, and the two attain maximal order simultaneously.
ER  -

Abstract

Related

Cite