The motivic zeta function and its smallest poles

TL;DR

This paper derives a new formula for the motivic zeta function of a regular function on a complex algebraic variety, relates it to embedded resolutions, and explores implications for its poles.

Contribution

It provides a formula for the motivic zeta function over the Grothendieck ring and analyzes the structure of jet spaces and poles.

Findings

Formula for motivic zeta function in terms of embedded resolution

Partition of jet spaces into locally closed subsets

Insights into the poles of the motivic zeta function

Abstract

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f=0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension the round up of dn/2. Finally, we look at the consequences for the poles of the motivic zeta function.