Monodromy Conjecture for log generic polynomials

TL;DR

This paper proves the monodromy conjecture for log generic polynomials and their products, establishing a connection between motivic zeta functions and monodromy in algebraic geometry.

Contribution

It demonstrates the validity of the monodromy conjecture for tuples involving log generic polynomials and their products, under certain genericity conditions.

Findings

Monodromy conjecture holds for log generic polynomials and their products.

The stronger conjecture relating motivic zeta functions and Bernstein-Sato ideals is proven for log very-generic cases.

Results extend the understanding of monodromy in the context of algebraic hypersurfaces.

Abstract

A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational modification of $\mathbb{P}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product $f=f_1\ldots f_p$ of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the tuple $(f_1,\ldots,f_p,g)$ and for the product $fg$, if $g$ is log very-generic.