有限域上非线性一元多项式族的因式分解模式
Factorization patterns on nonlinear families of univariate polynomials over a finite field
摘要：
我们估计了具有因子分解模式$\boldsymbolλ:=1^{λ_1}2^{λ_2}\cdots r^{λ_r}$的$\mathbb{F}_q[T]$的一元多项式的非线性族$\mathcal{A}$上的元素个数$|\mathcal{A}_{\boldsymbolλ}|$。我们证明了$|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$，其中$\mathcal{T}(\boldsymbolλ)$是具有循环模式的$r$元素的对称群的元素的比例$\boldsymbolλ$和$m$是$\mathcal{A}$的余维。我们提供明确的上限的常数的基础$\mathcal{O}$-符号的$\boldsymbolλ$和$\mathcal{A}$“好”的行为。我们还应用这些结果来分析的平均情况下的复杂性的经典因式分解算法限制在$\mathcal{A}$，表明它的行为一样好，在一般情况下。
英文摘要：
We estimate the number $|\mathcal{A}_{\boldsymbolλ}|$ of elements on a nonlinear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $r$ having factorization pattern $\boldsymbolλ:=1^{λ_1}2^{λ_2}\cdots r^{λ_r}$. We show that $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbolλ)$ is the proportion of elements of the symmetric group of $r$ elements with cycle pattern $\boldsymbolλ$ and $m$ is the codimension of $\mathcal{A}$. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbolλ$ 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.
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