Bernstein-Sato polynomials are fundamental in D-module theory. For example, they are the main finiteness ingredient in the construction of nearby cycles.

We will present a positive characteristic analogue of the Bernstein-Sato polynomials. After diving in the world of characteristic p D-modules, we shall consider how our construction varies with the prime p. This turns out to be related to questions of Hodge theory and Poisson homology.