of an embedded resolution $\varphi: \tilde X \rightarrow {\mathbb C}^{g+1}$ of $f$: they are zeros or poles of the monodromy zeta functions $$Z_{f,y}^{mon} (t)= \prod_{j \in J}(1-t^{N_j})^{\chi(E^{\circ}_j\cap \varphi^{-1}(y))}$$ for points $y \in \{f=0\}$. Both A’Campo formulas can be generalized in a straightforward way to ideals and polynomials, respectively, defining a subscheme $Y$ of a general variety $X$ with $\text{Sing}(X) \subset Y$.

We will apply formula  to a specific point in the exceptional divisor $E'$ that we define by means of a *generic embedding surface* of $Y$. For every set $(\lambda_2, \ldots, \lambda_g) $ of $g-1$ non-zero complex numbers, we introduce an affine scheme $S(\lambda_2,\ldots, \lambda_g)$ in ${\mathbb C}^{g+1}$ given by the equations $$\left\{ 
    \begin{array}{c c l l l}
    f_1& + & \lambda_2f_2  &  = 0 \\
    f_2& + & \lambda_3f_3 &= 0  \\
    & \vdots & &\\
    f_{g-1} & +  & \lambda_gf_g & = 0.\\
    \end{array}
    \right.$$ Every such scheme contains $Y$ as a Cartier divisor defined by one of the equations $f_i = 0$. For *generic* coefficients $(\lambda_2,\ldots, \lambda_g)$, the scheme $S(\lambda_2,\ldots, \lambda_g)$ is a normal surface which is smooth outside the origin. If we denote by $S'$ the strict transform of such a generic embedding surface $S := 