Bernstein polynomial and Tjurina number

The Bernstein polynomial of an isolated hypersurface singularity has subtle relations with the spectral numbers and the Tjurina number. To study these relations we use the Gauss-Manin connection, Malgrange's description of the Bernstein polynomial and ideas of M. Saito. A general discussion of mu-constant families leads to manageable methods for explicit calculations, which we use on a number of examples. We introduce a matrix which determines simultaneously the Bernstein polynomial and the Tjurina number.