Factorization patterns on nonlinear families of univariate polynomials over a finite field

We estimate the number $|\mathcal{A}_{\boldsymbol\lambda}|$ of elements on a nonlinear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $r$ having factorization pattern $\boldsymbol\lambda:=1^{\lambda_1}2^{\lambda_2}\cdots r^{\lambda_r}$. We show that $|\mathcal{A}_{\boldsymbol\lambda}|= \mathcal{T}(\boldsymbol\lambda)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbol\lambda)$ is the proportion of elements of the symmetric group of $r$ elements with cycle pattern $\boldsymbol\lambda$ and $m$ is the codimension of $\mathcal{A}$. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbol\lambda$ and $\mathcal{A}$ with "good" behavior. We also apply these results to analyze the average--case complexity of the classical factorization algorithm restricted to $\mathcal{A}$, showing that it behaves as good as in the general case.