Bernstein-Sato Theory in Positive Characteristic

Abstract

Given a holomorphic function f, its Bernstein-Sato polynomial is a classical invariant that detects the singularities of the zero locus of f in very subtle ways; for example, its roots recover the log-canonical threshold of f and the eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this thesis we continue the work of Bitoun and Mustață to develop an analogue of this invariant in positive characteristic. More concretely, we develop a notion of Bernstein-Sato polynomial for arbitrary ideals (which, over the complex numbers, was done by Budur, Mustață and Saito), we show that its roots are always rational and negative and that they encode some information about the F-jumping numbers. We also prove that for monomial ideals we can recover the roots of the classical Bernstein-Sato polynomial from this characteristic-p version.