Denef-Loeser zeta functions of suspensions and L\^e-Yomdin singularities

Abstract

The holomorphy conjecture for suspensions of plane curve singularities and the holomorphy and monodromy conjectures for L\^e-Yomdin singularities of surfaces are proved. The first part of this paper provides formul{\ae} for the motivic and topological zeta functions for a family of hypersurfaces, including the suspensions by an arbitrary number of points and which are more general than Thom-Sebastiani type. These formulae generalize and are inspired by the description of the topological and the 2-twisted topological zeta functions of suspensions by 2 points of hypersurfaces, due to the first named author, Cassou-Nogu\`es, Luengo and Melle. The new general formul{\ae} deal with arbitrary values of the twisting parameter. An interesting feature of these general formul{\ae} is the appearance of values of the Jordan's totient function as coefficients of the topological and the twisted topological zeta functions of some auxiliary hypersurfaces of smaller dimension.