The Bernstein-Sato $b$-function of the Vandermonde determinant

TL;DR

This paper advances the understanding of the Bernstein-Sato $b$-function for the Vandermonde determinant, proving conjectures for Coxeter arrangements, establishing root symmetry, and proposing a formula for the $b$-function.

Contribution

It provides bounds, symmetry results, and a conjectured formula for the $b$-function of the Vandermonde determinant, extending the theory of singularities and hyperplane arrangements.

Findings

Proved the Strong Monodromy Conjecture for finite Coxeter arrangements.

Established the symmetry of roots of the $b$-function about -1.

Proposed a conjectural formula for the $b$-function of the Vandermonde determinant.

Abstract

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the Vandermonde determinant $\xi$. We use a result of Opdam to produce a lower bound for the $b$-function of $\xi$. This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case. In our second set of results, we show the duality of two $\mathcal{D}$-modules, and conclude that the roots of the $b$-function of $\xi$ are symmetric about $-1$. We then use some results about jumping coefficients to prove an upper bound for the $b$-function of $\xi$, and finally we conjecture a formula for the $b$-function of $\xi$.