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

TL;DR

This paper estimates the distribution of factorization patterns in nonlinear families of univariate polynomials over finite fields, providing explicit bounds and analyzing the average-case complexity of factorization algorithms within these families.

Contribution

It introduces a precise asymptotic formula for counting polynomials with given factorization patterns in nonlinear polynomial families over finite fields.

Findings

Asymptotic count of polynomials with specific factorization patterns

Explicit bounds for constants in the asymptotic formula

Average-case complexity of factorization algorithms remains optimal

Abstract

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.