complex varieties $K_0(\text{Var}_{\mathbb C})$, and $\mathcal{M}_{\mathbb C}$ is the localization of $K_0(\text{Var}_{\mathbb C})$ with respect to ${\mathbb L}$. The precise definition of the Grothendieck ring of complex varieties can be found for instance in [@MVV]. In the *global version* of the motivic zeta function, we replace $[E^{\circ}_I\cap\varphi^{-1}(0)]$ by $[E^{\circ}_I]$. From this expression, it is immediate that both the local and the global motivic zeta function are rational functions in $T$, and that all candidate poles are of the form ${\mathbb L}^{\frac{\nu_j}{N_j}}$ for some $j \in J$. In concrete examples ‘most’ of these candidate poles cancel; a phenomenon that the monodromy conjecture tries to explain.

It is worth mentioning that the above expression (for one polynomial) is not the original definition given by Denef and Loeser in [@DL2]; they introduced the motivic Igusa zeta function for a polynomial $f$ using the *jet schemes* of $\{f = 0\}$. However, in the same article, they showed the equivalence between both expressions. Similarly, one can write the motivic zeta function associated with a general ideal ${\mathcal I}$ in terms of the jet schemes of its corresponding scheme $V({\mathcal I})$. In fact, this is the definition used to compute the motivic zeta function of a 