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On the degrees of polynomial divisors over finite fields

Published online by Cambridge University Press:  19 May 2016

ANDREAS WEINGARTNER
ANDREAS WEINGARTNER*
Affiliation:
Southern Utah University, Cedar City, Utah 84720, U.S.A. e-mail: weingartner@suu.edu
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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 cqn−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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 469 - 487
Copyright
Copyright © Cambridge Philosophical Society 2016 

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