Iteration of Quadratic Polynomials Over Finite Fields

D. R. Heath-Brown

Iteration of Quadratic Polynomials Over Finite Fields

Number Theory 2017-01-18 v2

Authors: D. R. Heath-Brown

Abstract

For a finite field of odd cardinality $q$ , we show that the sequence of iterates of $aX^2+c$ , starting at $0$ , always recurs after $O(q/\log\log q)$ steps. For $X^2+1$ the same is true for any starting value. We suggest that the traditional "Birthday Paradox" model is inappropriate for iterates of $X^3+c$ , when $q$ is 2 mod 3.

Keywords

finite fields quadratic polynomials iteration recurrence birthday paradox number theory

Cite

@article{arxiv.1701.02707,
  title  = {Iteration of Quadratic Polynomials Over Finite Fields},
  author = {D. R. Heath-Brown},
  journal= {arXiv preprint arXiv:1701.02707},
  year   = {2017}
}

Comments

This revision acknowledges prior work by Juul, Kurlberg, Madhu and Tucker, and by Shao, proving results that are closely related to those of the current paper

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