GRASP seminar: Introduction to D-modules via Bernstein-Sato polynomials

Seminar | February 8 | 2-3 p.m. | 732 Evans Hall

 Aaron Brookner, UCB

 Department of Mathematics

Given a real polynomial $p$ in $n$-variables, we define a family of distributions over $\{\lambda \in \mathbb C | \text {Re}\lambda >0\}$, generalizing the Γ-function. We claim that it is possible to analytically continue this family, exactly the same way as is traditionally done for the Γ-function, using the existence of the so-called Bernstein-Sato polynomial, $b_p$, of $p$.

We compute some of examples of these polynomials, then show that their existence is equivalent to a purely module-theoretical question. This takes us into the theory of holonomic $\mathcal D$-modules on $\mathbb A^n$, which we shall use as a diving board into the theory of $\mathcal D$-modules and to solve the original analytic question.