Seminário das Sextas, 04/08

Bernstein-Sato polynomials and the division of distributions

We will discuss some basic properties of left modules over the Weyl algebra ('algebraic D-modules on an affine space') up to the J. Bernstein's inequality and the notion of a holonomic module. We'll proceed to prove the existence of Bernstein-Sato polynomials and employ them to analytically continue complex powers of polynomials. As an application, a Hironaka-free solution to the problem of division of distributions by polynomials will follow. I will conclude with a remark about how all this provides a purely algebraic proof of theorems of Malgrange-Ehrenpreis and Lojasiewicz-Hormander about the existence of fundamental solutions of linear PDEs with constant coefficients.