The distribution of factorization patterns on linear families of polynomials over a finite field

We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern $\boldsymbol{\lambda}:=1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}$. We show that $|\mathcal{A}_{\boldsymbol{\lambda}}|= \mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbol{\lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $\boldsymbol{\lambda}$ and $m$ is the codimension of $\mathcal{A}$. Furthermore, if the family $\mathcal{A}$ under consideration is "sparse", then $|\mathcal{A}_{\boldsymbol{\lambda}}|= \mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $\mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbol{\lambda}$ and $\mathcal{A}$ with "good" behavior. Our approach reduces the question to estimate the number of $\mathbb{F}_q$--rational points of certain families of complete intersections defined over $\mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $\mathbb{F}_q$--rational points are established.