affine space ${\mathbb C}^{n+1}$. Assume that $Y$ contains the origin.

An important notion needed to introduce the monodromy conjecture for ${\mathcal I}$ is a *principalization (or log-principalization, log-resolution, monomialization) of an ideal*. If ${\mathcal I}= (f)$ is generated by a single polynomial, one can desingularize the hypersurface $Y = \{f = 0\}$ in ${\mathbb C}^{n+1}$ by repeatedly blowing up along well-chosen centers. This construction is called an *embedded resolution (of singularities)* of $\{f=0\}$ and always exists by Hironaka’s theorem [@Hi]. A sequence of blow-ups can also be used to transform a general ideal ${\mathcal I}= (f_1, \ldots, f_r)$ into a locally principal and monomial ideal. More formally, a *principalization* of ${\mathcal I}$ is a proper birational morphism $\varphi: \tilde{X} \rightarrow {\mathbb C}^{n+1}$ from a smooth variety $\tilde{X}$ to ${\mathbb C}^{n+1}$ such that the total transform $\varphi^{\ast}{\mathcal I}$ is a locally principal and monomial ideal with support a simple normal crossings divisor, and such that the exceptional locus (or exceptional divisor) of $\sigma$ is contained in the support of $\varphi^{\ast}{\mathcal I}$. The existence of a principalization is again a result of Hironaka [@Hi].

The motivic Igusa zeta function associated with ${\mathcal I}$ can be expressed in terms of a principalization $\varphi: 