# Splicing motivic zeta functions

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## Abstract

The first part of the paper discusses how to lift the splicing formula of Némethi and Veys 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 Némethi 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 Némethi 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 illustrate this with some examples.

## Keywords

Splicing Motivic zeta functions Monodromy conjecture## Mathematics Subject Classification

Primary 14E18 Secondary 14H50 32S05## Notes

### Acknowledgments

I am very grateful to Wim Veys and Johannes Nicaise for their valuable suggestions. I am also very grateful for the suggestions made by the referees.

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