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MULTIPLICATIVE ORDERS IN ORBITS 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:  23 October 2017

IGOR E. SHPARLINSKI Open the ORCID record for IGOR E. SHPARLINSKI [Opens in a new window]
IGOR E. SHPARLINSKI*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia e-mail: igor.shparlinski@unsw.edu.au
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Abstract

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We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

MSC classification

Primary: 11T06: Polynomials
Secondary: 37P05: Polynomial and rational maps 37P25: Finite ground fields
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 487 - 493
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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