AI 中文总结
本文确定通用对称Hankel矩阵行列式的整体Bernstein--Sato多项式，通过近旁循环、阿基米德zeta积分及Gauss--Radau参数化等方法，证明了极点与根的关系及单值类的唯一性。
AI 中文摘要
设 $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ 为大小为 $m$ 的通用对称Hankel矩阵的行列式。我们确定其整体Bernstein--Sato多项式。证明结合了相应割线超曲面的近旁循环与正定Hankel矩阵锥上的显式阿基米德zeta积分。Gauss--Radau参数化将该积分化为Morris积分，进而化为Gamma函数的商。其本原极点由支撑在锥顶点任意近处的测试函数检测。Lichtin定理应用于割线层状对数解消，确定了这些极点与Bernstein--Sato多项式根之间的整数平移。近旁循环的相对 $D[s]$-格实现，连同相应交复形的单性，给出每个单值类中的唯一性和重数一。矩阵大小上的归纳由割线簇的局部乘积结构得出。
英文摘要
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.
Comments33 pages
