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ITERATION OF QUADRATIC POLYNOMIALS OVER FINITE FIELDS

Part of: Arithmetic and non-Archimedean dynamical systems Polynomials and matrices Computational number theory Finite fields and commutative rings (number-theoretic aspects) Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 November 2017

D. R. Heath-Brown
D. R. Heath-Brown*
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K. email [email protected]
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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 $O(q/\text{log}\log q)$ 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.

MSC classification

Primary: 11T06: Polynomials
Secondary: 11C08: Polynomials 11G20: Curves over finite and local fields 11Y05: Factorization 37P05: Polynomial and rational maps 37P25: Finite ground fields
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 1041 - 1059
Copyright
Copyright © University College London 2017 

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References

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