ON THE HOLOMORPHY CONJECTURE FOR IGUSAS LOCAL ZETA-FUNCTION

To a polynomial f over a p-adic field K and a character chi of the gro up of units of the valuation ring of K one associates Igusa's local ze ta function Z(s, f, chi), which is a meromorphic function on C. Severa l theorems and conjectures relate the poles of Z(s, f, chi) to the mon odromy of f; the so-called holomorphy conjecture states roughly that i f the order of chi does not divide the order of any eigenvalue of mono dromy of f, then Z(s, f, chi) is holomorphic on C. We prove mainly tha t if the holomorphy conjecture is true for f(x(1),...,x(n-1)), then it is true for f(n(1),...,x(n-1)) + x(n)(k) with k greater than or equal to 3, and we give some applications.