The Bernstein-Sato Polynomial of the Hankel Determinant
Abstract
Let fm(x0,…,x2m−2) =det(xi+j)0≤ i,j≤ m−1 be the determinant of the generic symmetric Hankel matrix of size m. We determine its global Bernstein--Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the cone of positive definite Hankel matrices. A Gauss--Radau parametrization reduces this integral to a Morris integral and hence to a quotient of Gamma functions. Its primitive poles are detected by test functions supported arbitrarily close to the vertex of the cone. Lichtin's theorem, applied to a secant-stratum log resolution, fixes the integer shift between these poles and the roots of the Bernstein--Sato polynomial. A relative D[s]-lattice realization of nearby cycles, together with the simplicity of the corresponding intersection complexes, gives uniqueness and multiplicity one in every monodromy class. The induction in the matrix size follows from the local product structure of secant varieties.
Related
- A Polynomial PDE Criterion for the −n/d Root of the Bernstein-Sato polynomial of Homogeneous Ideals
- Zero loci of Bernstein-Sato ideals are not necessarily linear
- A combinatorial criterion for detecting roots of Bernstein-Sato polynomials
- On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals
- Bernstein--Sato ideals might not be linear
BibTeX
@misc{lin2026bernsteinsato,
title = {{The Bernstein-Sato Polynomial of the Hankel Determinant}},
author = {Xinyu Lin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19763},
eprint = {2609.19763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19763},
}
RIS
TY - UNPB
TI - The Bernstein-Sato Polynomial of the Hankel Determinant
AU - Xinyu Lin
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19763
UR - https://arxiv.org/abs/2609.19763
AB - Let f_m(x_0,…,x_2m−2) =det(x_i+j)_0≤ i,j≤ m−1 be the determinant of the generic symmetric Hankel matrix of size m. We determine its global Bernstein--Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the cone of positive definite Hankel matrices. A Gauss--Radau parametrization reduces this integral to a Morris integral and hence to a quotient of Gamma functions. Its primitive poles are detected by test functions supported arbitrarily close to the vertex of the cone. Lichtin's theorem, applied to a secant-stratum log resolution, fixes the integer shift between these poles and the roots of the Bernstein--Sato polynomial. A relative D[s]-lattice realization of nearby cycles, together with the simplicity of the corresponding intersection complexes, gives uniqueness and multiplicity one in every monodromy class. The induction in the matrix size follows from the local product structure of secant varieties.
ER -