On the Holomorphy Conjecture for Igusa's Local Zeta Function

To a polynomial f over a p{adic eld K and a character of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s; f;), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s; f;) to the monodromy of f; the so{called holomorphy conjecture states roughly that if the order of does not divide the order of any eigenvalue of monodromy of f, then Z(s; f;) is holomorphic on C. We prove mainly that if the holomorphy conjecture is true for f(x 1 ;. .. ; x n?1), then it is true for f(x 1 ;. .. ; x n?1) + x k n with k 3, and we give some applications.