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February  2020, 40(2): 1065-1073. doi: 10.3934/dcds.2020070

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

Received  May 2019 Revised  September 2019 Published  November 2019

Fund Project: 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.

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: Polynomial iterations, polynomials semigroups, multiplicative subgroup, finite fields, unlikely intersection.
Mathematics Subject Classification: Primary: 11D79, 11T06, 37P05, 37P25.
Citation: László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070
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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[4]

M.-C. Chang, Polynomial iteration in characteristic $p$, J. Functional Analysis, 263 (2012), 3412-3421.  doi: 10.1016/j.jfa.2012.08.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. C. Mason, Diophantine Equations over Function Fields, London Math. Soc., Lecture Note Series, 96, Cambridge University Press, 1984. doi: 10.1017/CBO9780511752490.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Ostafe, L. Pottmeyer and I. E. Shparlinski, Perfect powers in value sets and orbits of polynomials, preprint, arXiv: 1907.12057. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

I. E. Shparlinski, Multiplicative orders in orbits of polynomials over finite fields, Glasg. Math. J., 60 (2018), 487-493.  doi: 10.1017/S0017089517000222.  Google Scholar

[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.  Google Scholar

[25]

I. Vyugin, On the bound of inverse images of a polynomial map, preprint, arXiv: 1811.08930. Google Scholar

[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.  Google Scholar

[27]

J. Xie, The dynamical Mordell-Lang conjecture for polynomial endomorphisms of the affine plane, Astérisque, 394 (2017), 1–110.  Google Scholar

[28] U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Princeton Univ. Press, Princeton, NJ, 2012.   Google Scholar

show all references

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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[4]

M.-C. Chang, Polynomial iteration in characteristic $p$, J. Functional Analysis, 263 (2012), 3412-3421.  doi: 10.1016/j.jfa.2012.08.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. C. Mason, Diophantine Equations over Function Fields, London Math. Soc., Lecture Note Series, 96, Cambridge University Press, 1984. doi: 10.1017/CBO9780511752490.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Ostafe, L. Pottmeyer and I. E. Shparlinski, Perfect powers in value sets and orbits of polynomials, preprint, arXiv: 1907.12057. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

I. E. Shparlinski, Multiplicative orders in orbits of polynomials over finite fields, Glasg. Math. J., 60 (2018), 487-493.  doi: 10.1017/S0017089517000222.  Google Scholar

[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.  Google Scholar

[25]

I. Vyugin, On the bound of inverse images of a polynomial map, preprint, arXiv: 1811.08930. Google Scholar

[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.  Google Scholar

[27]

J. Xie, The dynamical Mordell-Lang conjecture for polynomial endomorphisms of the affine plane, Astérisque, 394 (2017), 1–110.  Google Scholar

[28] U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Princeton Univ. Press, Princeton, NJ, 2012.   Google Scholar
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