Almost non-degenerate functions and a Zariski pair of links

TL;DR

This paper extends the Varchenko formula to certain degenerate functions and provides an example of hypersurfaces with identical invariants but different geometric structures.

Contribution

It generalizes the Varchenko formula to functions with Newton degenerate faces and constructs a Zariski pair of links with identical invariants but different tangent cones.

Findings

Extended Varchenko formula to Newton degenerate functions

Constructed a Zariski pair of hypersurfaces with same invariants

Demonstrated differences in tangent cones despite identical zeta functions

Abstract

Let $f(\mathbf z)$ be an analytic function defined in the neighborhood of the origin of $\mathbb C^n$ which have some Newton degenerate faces. We generalize the Varchenko formula for the zeta function of the Milnor fibration of a Newton non-degenerate function $f$ to this case. As an application, we give an example of a pair of hypersurfaces with the same Newton boundary and the same zeta function with different tangent cones.