| Summary: | The Bernstein-Sato polynomial, or the b-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general.
Chapter 1 gives an introduction to the subject, explaining the framework of D-modules, from which the existence of b-functions comes. We also give some basic examples and properties. Lastly, we define the related singularity invariants of Jumping coefficients, Milnor monodromy, and Zeta functions, which are often intertwined with b-function computations.
In Chapter 2, we compute the Bernstein-Sato polynomial of f, a function which given a pair (M, v) in X = Mn(C) x Cn tests whether v is a cyclic vector for M. The proof includes a description of shift operators corresponding to the Calogero-Moser operator Lk in the rational case.
In Chapter 3, we prove a few different results towards computing the b-function of the Vandermonde determinant zeta. We use a result of Opdam to produce a lower bound for the b-function of zeta. This bound proves a conjecture of Budur, Mustataˇ, and Teitler for the case of finite Coxeter hyperplane arrangements, proving the Strong Monodromy Conjecture in this case.
In the second set of results, we show the duality of two D-modules, and conclude that the roots of the b-function of zeta are symmetric about -1. We then use some results about jumping coefficients to prove an upper bound for the b-function of zeta, and finally we conjecture a formula for the b-function of zeta.
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