\tilde{X} \rightarrow {\mathbb C}^{n+1}$ of ${\mathcal I}$ as follows. Let $E_j$ for $j \in J$ be the irreducible components (with their reduced scheme structure) of the total transform $\varphi^{-1}(Y)$. Among these, the components of the exceptional divisor are called the *exceptional varieties*; the other components are components of the *strict transform* of $Y$. Denote by $N_j$ the multiplicity of $E_j$ in the divisor on $\tilde{X}$ of $\varphi^{\ast}{\mathcal I}$, that is, the divisor of $\varphi^{\ast}{\mathcal I}$ is given by $\sum_{i \in J}N_jE_j$. Similarly, let $\nu_j-1$ be the multiplicity of $E_j$ in the divisor on $\tilde{X}$ of $\varphi^{\ast}(dx_1 \wedge \ldots \wedge dx_n)$. The numbers $(N_j,\nu_j)$ for $j \in J$ are called the *numerical data* of the principalization. For every subset $I \subset J$, we also define $E_I^\circ := (\cap_{i \in I} E_i) \backslash (\cup_{l \not \in I}E_l)$. In terms of this notation, the *local motivic Igusa zeta function* associated with the ideal ${\mathcal I}$ (or with the scheme $Y$) is given by $$Z^{mot}_{{\mathcal I}}(T) = {\mathbb L}^{-(n+1)}\sum_{I\subset J} [E^{\circ}_I \cap\varphi^{-1}(0)]\prod_{i \in I} \frac{({\mathbb L}-1){\mathbb L}^{-\nu_i}T^{N_i}}{1-{\mathbb L}^{-\nu_i}T^{ N_i}} \in \mathcal{M}_{{\mathbb C}}[[T]].$$ Here, $[E^{\circ}_I\cap\varphi^{-1}(0)]$ and ${\mathbb L}:= [{\mathbb C}]$ are the class of $E^{\circ}_I\cap\varphi^{-1}(0)$ and the affine line, respectively, in the *Grothendieck ring* of 