On the degrees of polynomial divisors over finite fields

TL;DR

This paper derives asymptotic formulas for the proportion of polynomials over finite fields with divisors of all degrees below their degree, revealing connections to permutation cycle structures as the field size grows.

Contribution

It provides new asymptotic estimates for the distribution of polynomial divisors over finite fields, extending previous results and linking to permutation cycle structures.

Findings

Proportion of polynomials with divisors of every degree is approximately c_q n^{-1}.

Asymptotic formula for polynomials with no large gaps in divisor degrees.

Results align with permutation cycle estimates as q approaches infinity.

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 $q \to \infty$, these results coincide with corresponding estimates related to the cycle structure of permutations.