On the degrees of polynomial divisors over finite fields

Source abstract

Abstract We show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n , is given by c q n −1 + O ( n −2 ). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m . To that end, we first derive an improved estimate for the proportion of polynomials of degree n , all of whose non-constant divisors have degree greater than m . In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permutations.