The Bernstein-Sato Polynomial of the Hankel Determinant

Abstract

Let $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ 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.