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: April 30, 2019

Revised: August 31, 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

Abstract Full Text(HTML) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	R. Benedetto, D. Ghioca, P. Kurlberg and T. Tucker, A case of the dynamical Mordell-Lang conjecture, Math. Ann., 352 (2012), 1-26. doi: 10.1007/s00208-010-0621-4.
[2]	A. Bérczes, A. Ostafe, I. E. Shparlinski and J. H. Silverman, Multiplicative dependence among iterated values of rational functions modulo finitely generated groups, Intern. Math. Res. Notices (to appear).
[3]	J. Cahn, R. Jones and J. Spear, Powers in orbits of rational functions: Cases of an arithmetic dynamical Mordell–Lang conjecture, Canad. Math. J., 71 (2019), 773-817. doi: 10.4153/CJM-2018-026-x.
[4]	M.-C. Chang, Polynomial iteration in characteristic $p$, J. Functional Analysis, 263 (2012), 3412-3421. doi: 10.1016/j.jfa.2012.08.018.
[5]	M.-C. Chang, Expansions of quadratic maps in prime fields, Proc. Amer. Math. Soc., 142 (2014), 85-92. doi: 10.1090/S0002-9939-2013-11740-5.
[6]	E. Chen, Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures, Proc. Amer. Math. Soc., 146 (2018), 4189-4198. doi: 10.1090/proc/14115.
[7]	J. Cilleruelo, M. Z. Garaev, A. Ostafe and I. Shparlinski, On the concentration of points of polynomial maps and applications, Math. Z., 272 (2012), 825-837. doi: 10.1007/s00209-011-0959-7.
[8]	R. Dvornicich and U. Zannier, Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J., 139 (2007), 527-554. doi: 10.1215/S0012-7094-07-13934-6.
[9]	D. Ghioca, T. Tucker and M. Zieve, Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture, Invent. Math., 171 (2008), 463-483. doi: 10.1007/s00222-007-0087-5.
[10]	D. Ghioca, T. Tucker and M. Zieve, Linear relations between polynomial orbits, Duke Math. J., 161 (2012), 1379-1410. doi: 10.1215/00127094-1598098.
[11]	H. Krieger, A. Levin, Z. Scherr, T. Tucker, Y. Yasufuku and M. E. Zieve, Uniform boundedness of S-units in arithmetic dynamics, Pacific J. Math., 274 (2015), 97-106. doi: 10.2140/pjm.2015.274.97.
[12]	R. C. Mason, Diophantine Equations over Function Fields, London Math. Soc., Lecture Note Series, 96, Cambridge University Press, 1984. doi: 10.1017/CBO9780511752490.
[13]	H. Niederreiter and A. Winterhof, Multiplicative character sums for nonlinear recurring sequences, Acta Arith., 111 (2004), 299-305. doi: 10.4064/aa111-3-6.
[14]	H. Niederreiter and A. Winterhof, Exponential sums for nonlinear recurring sequences, Finite Fields Appl., 14 (2008), 59-64. doi: 10.1016/j.ffa.2006.09.010.
[15]	A. Ostafe, On roots of unity in orbits of rational functions, Proc. Amer. Math. Soc., 145 (2017), 1927-1936. doi: 10.1090/proc/13433.
[16]	A. Ostafe, Polynomial values in affine subspaces over finite fields, J. d'Analyse Math., 138 (2019), 49-81. doi: 10.1007/s11854-019-0021-y.
[17]	A. Ostafe, L. Pottmeyer and I. E. Shparlinski, Perfect powers in value sets and orbits of polynomials, preprint, arXiv: 1907.12057.
[18]	A. Ostafe and M. Sha, On the quantitative dynamical Mordell-Lang conjecture, J. Number Theory, 156 (2015), 161-182. doi: 10.1016/j.jnt.2015.04.011.
[19]	A. Ostafe and I. E. Shparlinski, Orbits of algebraic dynamical systems in subgroups and subfields', in Number theory–-Diophantine Problems, Uniform Distribution and Applications, 347–368, Springer, Cham, 2017.
[20]	A. Ostafe and M. Young, On algebraic integers of bounded house and preperiodicity in polynomial semigroup dynamics, Trans. Amer. Math. Soc., 2019. doi: 10.1090/tran/7974.
[21]	O. Roche-Newton and I. E. Shparlinski, Polynomial values in subfields and affine subspaces of finite fields, Quart. J. Math., 66 (2015), 693-706. doi: 10.1093/qmath/hau032.
[22]	I. E. Shparlinski, Groups generated by iterations of polynomials over finite fields, Proc. Edinburgh Math. Soc., 59 (2016), 235-245. doi: 10.1017/S0013091515000097.
[23]	I. E. Shparlinski, Multiplicative orders in orbits of polynomials over finite fields, Glasg. Math. J., 60 (2018), 487-493. doi: 10.1017/S0017089517000222.
[24]	J. H. Silverman and B. Viray, On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture, Math. Res. Letters, 20 (2013), 547-566. doi: 10.4310/MRL.2013.v20.n3.a12.
[25]	I. Vyugin, On the bound of inverse images of a polynomial map, preprint, arXiv: 1811.08930.
[26]	J. Xie, Dynamical Mordell–Lang conjecture for birational polynomial morphisms on $\mathbb A^2$, Math. Ann., 360 (2014), 457-480. doi: 10.1007/s00208-014-1039-1.
[27]	J. Xie, The dynamical Mordell-Lang conjecture for polynomial endomorphisms of the affine plane, Astérisque, 394 (2017), 1–110.
[28]	U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Princeton Univ. Press, Princeton, NJ, 2012.