Deformations of polynomials and their zeta-functions

For an analytic in σ ε(ℂ 0) family P σ of polynomials in n variables a monodromy transformation h of the zero level set V σ ={P σ =0} for sufficiently small σ ≠ 0 is defined. The zeta-function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta-functions. We give some examples of computations with the use of this technique. © Springer Science+Business Media, Inc. 2007.