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Groups generated by iterations of polynomials over finite fields

Part of: Arithmetic and non-Archimedean dynamical systems Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  05 June 2015

Igor E. Shparlinski
Igor E. Shparlinski*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (igor.shparlinski@mq.edu.au)
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Abstract

Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

Keywords

finite fieldspolynomial mapssubgroupscharacter sums

MSC classification

Secondary: 11T06: Polynomials 37P05: Polynomial and rational maps
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 235 - 245
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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