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On the degrees of polynomial divisors over finite fields

  • ANDREAS WEINGARTNER (a1)
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.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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