The motivic zeta function and its smallest poles

Dirk Segers, Lise Van Proeyen, Willem Veys

Published 2007-10-31, updated 2012-09-17Version 2

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.