Splicing for motivic zeta functions

TL;DR

This paper extends the splicing formula to motivic zeta functions, introduces monodromic motivic zeta functions, and discusses their relation to the generalized monodromy conjecture, highlighting limitations and providing examples.

Contribution

It generalizes the splicing formula to the motivic setting and analyzes the properties and limitations of monodromic motivic zeta functions.

Findings

Splicing formula successfully extended to motivic zeta functions.

Monodromic motivic zeta functions cannot be fully captured by splice diagrams.

The generalized monodromy conjecture holds for motivic zeta functions but not for monodromic ones.

Abstract

We lift the splicing formula of N\'emethi and Veys, which deals with polynomials in two variables, to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of N\'emethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of N\'emethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We show some examples illustrating this.