$\tilde{M}^k_y$. For an ideal ${\mathcal I}$, Verdier considered the *normal cone* $C_Y{\mathbb C}^{n+1}$ of $Y = V({\mathcal I})$ in ${\mathbb C}^{n+1}$ defined as $$C_Y{\mathbb C}^{n+1} := \text{Spec}(\oplus_{k\geq 0}{\mathcal I}^k/{\mathcal I}^{k+1}),$$ and related to ${\mathbb C}^{\boldsymbol{\cdot}} \in D^b_c({\mathbb C}^{n+1})$ the *specialization complex* $\text{Sp}_Y({\mathbb C}^{\boldsymbol{\cdot}}) \in D^b_c(C_Y{\mathbb C}^{n+1})$ with a canonical monodromy transformation $\tilde{M}^k_y: \mathcal H^k(\text{Sp}_Y({\mathbb C}^{\boldsymbol{\cdot}}))_y \rightarrow \mathcal H^k(\text{Sp}_Y({\mathbb C}^{\boldsymbol{\cdot}}))_y$ for each $y \in C_Y{\mathbb C}^{n+1}\setminus Y$. The *(Verdier) monodromy eigenvalues* of ${\mathcal I}$ (or of $Y$) are the eigenvalues of these automorphisms, and we can collect them in zeta functions of monodromy similar to . Despite the fact that the specialization complex lives on the normal cone of $Y$ instead of on $Y$ itself, where the complex of nearby cycles lives, it turns out that these two definitions for the monodromy eigenvalues in the hypersurface case are equivalent.

Let us now take a look at how we can express these monodromy eigenvalues in terms of a principalization. In [@AC], A’Campo proved a formula for the monodromy zeta function  of a polynomial $f$ at a point $y \in Y = \{f = 0\}$ in terms of an embedded resolution of singularities. This formula was generalized to ideals in [@VV]. Later in 