space monomial curve $Y \subset {\mathbb C}^{g+1}$ in [@MVV]. We have chosen to introduce the motivic Igusa zeta function by means of a principalization in this article because the *monodromy eigenvalues* associated with ${\mathcal I}$ can also be expressed in terms of a principalization of ${\mathcal I}$.

Before going into the details of this, we first briefly discuss the original definition of monodromy eigenvalues for a polynomial $f\in {\mathbb C}[x_0,\ldots, x_n]$ with $f(0) = 0$. Consider $f$ as a map from ${\mathbb C}^{n+1} \rightarrow {\mathbb C}$ and take a point $y \in Y = \{f=0\}$. Let $B$ be a ball in ${\mathbb C}^{n+1}$ with center $y$ and $D$ a disc in ${\mathbb C}$ with center $0$. Denote by $D^{\ast} := D \setminus \{0\}$ the punctured disc. For $B$ and $D$ small enough, Milnor [@Mi] proved that the restriction $f\vert_{B \cap f^{-1}(D^{\ast})}: B \cap f^{-1}(D^{\ast}) \rightarrow  D^{\ast}$ is a smooth locally trivial fibration. We call this map the *(local) Milnor fibration* of $f$ at $y$ and its fiber the *(local) Milnor fiber* $F_y$ of $f$ at $y$. The lifting under the Milnor fibration of a loop in $D^{\ast}$ of the form $[0,1] \rightarrow D^{\ast}: t \mapsto \delta e^{2\pi it}$ yields a 