Uniform estimates for smooth polynomials over finite fields

Abstract: 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.