Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities

TL;DR

This paper proves the strong monodromy conjecture for certain hypersurfaces with weighted homogeneous isolated singularities, extending previous results and revealing cancellations that prevent counterexamples.

Contribution

It establishes the conjecture for specific classes of hypersurfaces by leveraging formulas for Newton-nondegenerate polynomials and known two-variable results.

Findings

The conjecture holds for reduced curves with weighted homogeneous singularities.

A cancellation phenomenon prevents potential counterexamples.

Relations between pole orders of zeta functions and Bernstein-Sato roots are shown.

Abstract

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous isolated singularities with $n$ at least $4$, we show that the strong monodromy conjecture for a defining polynomial $f$ of $Z$ follows from arxiv:1609.04801v1 using in the reduced curve case a formula of Denef and Loeser for Newton-nondegenerate polynomials of three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail. We also show the relation between the pole orders of topological zeta function and the root multiplicities of Bernstein-Sato polynomial in the case $Z$ has equimultiplicity and $Z_{\rm red}$ has only weighted homogeneous singularities with $n=3$ or $Z_{\rm red}$ has only homogeneous isolated singularities with $n>3$.