Deformations of polynomials and their zeta-functions

For an analytic in σ ∈ (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. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.