论文标题
重新审视平滑排列和多项式
Smooth permutations and polynomials revisited
论文作者
论文摘要
我们研究有限场上平滑排列和平滑多项式的计数。对于这两项计数，我们都证明了一个错误项的估计值，该估计与70多年前De Bruijn在整数设置中发现的误差术语相匹配。主要术语是通常的Dickman $ρ$函数，但随着论点的转移。 我们确定$ \ log（p_ {n，m}/ρ（n/m））$的数量级，其中$ p_ {n，m} $是$ n $元素上的排列，随机选择$ m $ -smoth的概率。 我们在多项式设置中发现了一个相变：$ \ mathbb {f} _q $的多项式$ n $的概率是$ m $ -s-s-morth在$ m \ of（3/2）\ log_q n $上更改其行为。
We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $ρ$ function, but with its argument shifted. We determine the order of magnitude of $\log(p_{n,m}/ρ(n/m))$ where $p_{n,m}$ is the probability that a permutation on $n$ elements, chosen uniformly at random, is $m$-smooth. We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree $n$ in $\mathbb{F}_q$ is $m$-smooth changes its behavior at $m\approx (3/2)\log_q n$.
