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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

REFERENCES

1. Akbary, A. and Ghioca, D., Periods of orbits modulo primes, J. Number Theory 129 (2009), 2831–2842.CrossRefGoogle Scholar
2. Bateman, P. T., Pomerance, C. and Vaughan, R. C., On the size of the coefficients of the cyclotomic polynomial, in Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. János Bolyai, vol. 34 (North-Holland, Amsterdam, 1984), 171–202.Google Scholar
3. Benedetto, R. L., Ghioca, D., Hutz, B., Kurlberg, P., Scanlon, T. and Tucker, T. J., Periods of rational maps modulo primes, Math. Ann. 355 (2013), 637–660.CrossRefGoogle Scholar
4. Chang, M.-C., On periods modulo p in arithmetic dynamics, C. R. Acad. Sci. Paris, Ser. I 353 (2015), 283–285.CrossRefGoogle Scholar
5. Chang, M.-C., D'Andrea, C., Ostafe, A., Shparlinski, I. E. and Sombra, M., Orbits of polynomial dynamical systems modulo primes, Proc. Amer. Math. Soc., (to appear).Google Scholar
6. D'Andrea, C., Ostafe, A., Shparlinski, I. E. and Sombra, M., Reduction modulo primes of systems of polynomial equations and algebraic dynamical systems, Preprint, 2015 (see http://arxiv.org/abs/1505.05814).Google Scholar
7. Dvornicich, R. and Zannier, U., Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J. 139 (2007), 527–554.CrossRefGoogle Scholar
8. Krick, T., Pardo, L. M. and Sombra, M., Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (2001), 521–598.CrossRefGoogle Scholar
9. Ostafe, A., Polynomial values in affine subspaces of finite fields, J. d'Analyse Math. (to appear).Google Scholar
10. Ostafe, A., On roots of unity in orbits of rational functions, Proc. Amer. Math. Soc. 145 (2017), 1927–1936.CrossRefGoogle Scholar
11. Ostafe, A. and Shparlinski, I. E., Orbits of algebraic dynamical systems in subfields and subgroups, in Number theory – diophantine problems, uniform distribution and applications; festschrift in honour of Robert F. Tichy's 60th Birthday (Elsholz, C. and Grabner, P., Editors) (Springer, 2017), 347–368.CrossRefGoogle Scholar
12. Roche-Newton, O. and Shparlinski, I. E., Polynomial values in subfields and affine subspaces of finite fields, Quart. J. Math. 66 (2015), 693–706.CrossRefGoogle Scholar
13. Shparlinski, I. E., Groups generated by iterations of polynomials over finite fields, Proc. Edinburgh Math. Soc. 59 (2016), 235–245.CrossRefGoogle Scholar
14. Silverman, J. H., Variation of periods modulo p in arithmetic dynamics, New York J. Math. 14 (2008), 601–616.Google Scholar