Unlikely intersections over finite fields: Polynomial orbits in small subgroups

László Mérai; Igor E. Shparlinski

doi:10.3934/dcds.2020070

Article Contents

2020, Volume 40, Issue 2: 1065-1073. Doi: 10.3934/dcds.2020070

This issue Previous Article Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case Next Article Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Unlikely intersections over finite fields: Polynomial orbits in small subgroups

László Mérai^1, , and
Igor E. Shparlinski^2,

1.
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria
2.
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

^* Corresponding author: László Mérai
^* Corresponding author: László Mérai

Received: May 2019

Revised: September 2019

Published: November 2019

The first author is supported by the Austrian Science Fund (FWF): Project P31762 and the second author is supported by the Australian Research Council Grants DP170100786 and DP180100201

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Abstract

We estimate the frequency of polynomial iterations which fall in a given multiplicative subgroup of a finite field of $ p $ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $ N $ elements in an orbit. We derive these from more general results about sequences of compositions on a fixed set of polynomials.

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Mathematics Subject Classification: Primary: 11D79, 11T06, 37P05, 37P25.

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