Uniform estimates for friable polynomials over finite fields

Abstract

We establish new estimates for the number of m-friable polynomials of degree n over a finite field Fq, where the main term involves the number of m-friable permutations on n elements. The range of m in our results is n ≥ m ≥ (1 + ε) log q n and is optimal as these numbers are not comparable for smaller m. For m ≥ 4 log q n the error term in our estimates decays faster than in all previous works.Our estimates imply that the probability that a random polynomial fn is m-friable is asymptotic to the probability that a random permutation πn is m-friable, uniformly for m ≥ (2 + ε) log q n as q n → ∞. This should be viewed as an unconditional analogue of a result of Hildebrand (1984), saying that under the Riemann Hypothesis, the probability that a number ≤ x is y-friable is of the order of magnitude of the Dickman ρ function at log x/ log y, uniformly for log y ≥ (2 + ε) log log x. As opposed to Hildebrand's result, we actually obtain an asymptotic result.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. This improves a previous estimate of Knopfmakher and Manstavičius (1997).