Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials
Abstract
In this paper, we relate multiplicities of Bernstein--Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain ``mod'' b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f, and we provide an algorithm to compute them. Second, we show that they can be approximated by the multiplicities of roots of power b-functions (the b-functions with respect to powers of f). Further, we give a sharp upper bound for the Hodge level of elements given by a certain sum of such multiplicities. Next, we give an effective asymptotic solution to the Gelfand problem by determining an explicit threshold after which every integer shift of a root of b_f(s) is a pole of the Archimedean zeta function of f. We also show that the order of these poles is equal to the nilpotency index of the logarithmic monodromy operator, which we further express as the limit of the multiplicities of roots of power b-functions. We define several filtrations, relating them to the weight and Hodge filtrations, based upon which we leave some open questions that we address in the affirmative in the case when f has a homogeneous isolated singularity, or it is a hyperplane arrangement, or it is a semi-invariant on a spherical variety. We give several immediate applications to our results, including a positive answer to a question of Torelli assuming the hypersurface has log canonical singularities: 1/f lies in the intersection complex of the hypersurface of f if and only if -1 is a simple root of b_f(s).
Related
- A combinatorial criterion for detecting roots of Bernstein-Sato polynomials
- Real polynomials with given multiplicities of real roots: Complete conjectural description of homology
- Affine Symmetries of Polynomial Roots with Multiplicities and Applications to Constacyclic Codes
- Analytic Criteria for Detecting Bernstein-Sato Roots
- Homological Trimming and Regularity of Filtrations via Local Obstruction Modules
BibTeX
@misc{lorincz2026filtrations,
title = {{Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials}},
author = {Andras Lorincz and Ruijie Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05215},
eprint = {2609.05215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05215},
}
RIS
TY - UNPB
TI - Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials
AU - Andras Lorincz
AU - Ruijie Yang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05215
UR - https://arxiv.org/abs/2609.05215
AB - In this paper, we relate multiplicities of Bernstein--Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain ``mod'' b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f, and we provide an algorithm to compute them. Second, we show that they can be approximated by the multiplicities of roots of power b-functions (the b-functions with respect to powers of f). Further, we give a sharp upper bound for the Hodge level of elements given by a certain sum of such multiplicities. Next, we give an effective asymptotic solution to the Gelfand problem by determining an explicit threshold after which every integer shift of a root of b_f(s) is a pole of the Archimedean zeta function of f. We also show that the order of these poles is equal to the nilpotency index of the logarithmic monodromy operator, which we further express as the limit of the multiplicities of roots of power b-functions. We define several filtrations, relating them to the weight and Hodge filtrations, based upon which we leave some open questions that we address in the affirmative in the case when f has a homogeneous isolated singularity, or it is a hyperplane arrangement, or it is a semi-invariant on a spherical variety. We give several immediate applications to our results, including a positive answer to a question of Torelli assuming the hypersurface has log canonical singularities: 1/f lies in the intersection complex of the hypersurface of f if and only if -1 is a simple root of b_f(s).
ER -