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.