etLet $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$. By Hironaka's desingularization theorem, there exists a birational map $\pi: X\to V$ such that the restriction map $\pi: X\backslash \pi^{-1}(Z_{sing})\to V\backslash Z_{sing}$ is an isomorphism; moreover $\pi^{-1}(Z)$ is the union of simple normal crossing divisors. For each divisor $E$, one can attach a pair of number $(N_E,n_E)$, where $N_E$ is the order of the function $\pi^*f$ along $E$ and $n_E$ is the order of $\pi^* \omega_V$ along $E$, where $\omega_V$ is the standard top form of $V$. These data are very important for the understanding of the singularity of $f$ and also have many applications in other fields.
I understand it is difficult to give explicit construction of resolution of singularity for each $f$. My question is, in the following examples, is there any references of the explicit desingularization and also the data $(N_E,n_E)$?
$(1)$. $f$ is the determinant on matrix space
$(2)$. $f=\sum_{i=1}^{n}x_iy_i$.
$(3)$. More generally, $f(A,B)=det(AB)$, where $A\in M_{nk}$ and $B\in M_{kn}$, $k\geq n$.
Thanks.
