Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals
Abstract
Let X be a smooth complex affine variety of dimension n, and let F=(f1,…,fr) be a tuple of nonzero regular functions on X such that f:=∏i=1r fi is not invertible. We study the zero loci of the Bernstein-Sato ideals BFa for nonnegative integral shifts a. For a fixed log resolution, every codimension-one irreducible component of Z(BFa) is a hyperplane of the form LE(s)+kE+c=0 with c a positive integer. We give a new proof of this result using localized maximal and minimal extensions of relative D-modules. We also prove that c≤ LE(a)+(n−1−δf)LE(1)−kE, where δf=min{n−1,αf} and αf is the minimal exponent of f. The problem of obtaining such an upper bound for arbitrary tuples (in particular, for r>1) was raised by Budur, van der Veer, and Van Werde, and the above inequality resolves it. To obtain the upper bound, we compare the diagonal slice of Z(BF1) with the root set of bf. A finite covering by translates, combined with diagonal specialization and the log-resolution description, shows that these sets have the same least and greatest points. Saito's root estimate at their common least point then yields the upper bound. We further establish a divisor-valued formulation of the local index comparison, recovering the detection of monodromy support via monodromy zeta functions and the multivariable A'Campo formula.
Related
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BibTeX
@misc{guo2026bounds,
title = {{Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals}},
author = {Wenzong Guo and Fanghan Xiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19869},
eprint = {2609.19869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19869},
}
RIS
TY - UNPB
TI - Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals
AU - Wenzong Guo
AU - Fanghan Xiang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.19869
UR - https://arxiv.org/abs/2609.19869
AB - Let X be a smooth complex affine variety of dimension n, and let F=(f_1,…,f_r) be a tuple of nonzero regular functions on X such that f:=∏_i=1^r f_i is not invertible. We study the zero loci of the Bernstein-Sato ideals B_F^a for nonnegative integral shifts a. For a fixed log resolution, every codimension-one irreducible component of Z(B_F^a) is a hyperplane of the form L_E(s)+k_E+c=0 with c a positive integer. We give a new proof of this result using localized maximal and minimal extensions of relative D-modules. We also prove that c≤ L_E(a)+(n−1−δ_f)L_E(1)−k_E, where δ_f=min{n−1,α_f} and α_f is the minimal exponent of f. The problem of obtaining such an upper bound for arbitrary tuples (in particular, for r>1) was raised by Budur, van der Veer, and Van Werde, and the above inequality resolves it. To obtain the upper bound, we compare the diagonal slice of Z(B_F^1) with the root set of b_f. A finite covering by translates, combined with diagonal specialization and the log-resolution description, shows that these sets have the same least and greatest points. Saito's root estimate at their common least point then yields the upper bound. We further establish a divisor-valued formulation of the local index comparison, recovering the detection of monodromy support via monodromy zeta functions and the multivariable A'Campo formula.
ER -