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Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case
Unlikely intersections over finite fields: Polynomial orbits in small subgroups
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 |
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.
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.
![]() ![]() |
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. |
[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.
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