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arXiv · 1503.01055

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

Asilata Bapat·Robin Walters
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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 $\xi$.

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Asilata Bapat, Robin Walters. 2015-03-03. The Bernstein-Sato $b$-function of the Vandermonde determinant. https://arxiv.org/abs/1503.01055

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