The Bernstein-Sato $b$-function of the Vandermonde determinant
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 $ξ$. We use a result of Opdam to produce a lower bound for the $b$-function of $ξ$. This bound proves a conjecture of Budur, Mustaţă, 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 $ξ$ are symmetric about $-1$. We then use some results about jumping coefficients to prove an upper bound for the $b$-function of $ξ$, and finally we conjecture a formula for the $b$-function of $ξ$.