The Bernstein-Sato $b$-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 $\xi$.