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

TL;DR

This paper proves the holomorphy and monodromy conjectures for certain singularities, providing new formulas for motivic and topological zeta functions that include twisted variants and involve Jordan's totient function.

Contribution

It generalizes formulas for zeta functions of hypersurface suspensions, extending previous work to arbitrary twisting parameters and broader classes of singularities.

Findings

Proved holomorphy conjecture for suspensions of plane curve singularities.

Established holomorphy and monodromy conjectures for L extsuperscript{e}-Yomdin singularities of surfaces.

Derived formulas involving Jordan's totient function for zeta functions of hypersurfaces.

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.