论文标题
在有限场上平滑多项式的均匀估计值
Uniform estimates for smooth polynomials over finite fields
论文作者
论文摘要
我们建立了有限字段$ \ mathbb {f} _q $的$ m $ -smooth多项式$ n $的新估算值，其中主术语涉及$ n $ elements上的$ m $ -smooth complutations。 我们的估计，这意味着$ n $的随机多项式为$ m $ -smooth的可能性是渐近的，因为$ n $元素上随机排列为$ m $ -smooth的可能性，均为$ m \ ge ge（2+ \ varepsilon）\ log_q n $ as $ q^n $ as $ q^n \。这应该被视为在整数环境中希尔德布兰德和赛亚斯作品的无条件类似物，这是假设的假设。此外，我们证明了$ m \ ge（2+ \ varepsilon）\ log_q n $ shark;这应该被视为Hildebrand猜想的（A）猜想的分辨率。 作为我们估计值的应用，我们确定了随机多项式最大质量因子的预期程度的渐近公式中的衰减速率。
We establish new estimates for the number of $m$-smooth polynomials of degree $n$ over a finite field $\mathbb{F}_q$, where the main term involves the number of $m$-smooth permutations on $n$ elements. Our estimates imply that the probability that a random polynomial of degree $n$ is $m$-smooth is asymptotic to the probability that a random permutation on $n$ elements is $m$-smooth, uniformly for $m\ge (2+\varepsilon)\log_q n$ as $q^n \to \infty$. This should be viewed as an unconditional analogue of works of Hildebrand and of Saias in the integer setting, which assume the Riemann Hypothesis. Moreover, we show that the range $m \ge (2+\varepsilon)\log_q n$ is sharp; this should be viewed as a resolution of a (polynomial analogue of a) conjecture of Hildebrand. As an application of our estimates, we determine the rate of decay in the asymptotic formula for the expected degree of the largest prime factor of a random polynomial.
