Resumen de On a theory of the b-function in positive characteristic

Thomas Bitoun

• We present a theory of the b-function (or Bernstein–Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p > 0. Its b-function b f is defined to be an ideal of the algebra of continuous k-valued functions on Zp. The zero-locus of the b-function is thus naturally interpreted as a subset of Zp, which we call the set of roots of b f . We prove that b f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta¸t˘a and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b f and relate it to the test ideals of f.