  We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of
polynomial maps of one complex variable with degree $d \geq 2$, and study the
map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d /
\mathfrak{S}_d$ which maps each $f \in \mathrm{MP}_d$ to the set of fixed-point
multipliers of $f$. We show that the local fiber structure of the map $\Phi_d$
around $\bar{\lambda} \in \widetilde{\Lambda}_d$ is completely determined by
certain two sets $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$ which are
subsets of the power set of $\{1,2,\ldots,d \}$. Moreover for any
$\bar{\lambda} \in \widetilde{\Lambda}_d$, we give an algorithm for counting
the number of elements of each fiber $\Phi_d^{-1}\left(\bar{\lambda}\right)$
only by using $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$. It can be
carried out in finitely many steps, and often by hand.
