Skip to main content Accessibility help

Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture

  • Jordan Cahn (a1), Rafe Jones (a1) and Jacob Spear (a1)

Let $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$ , we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$ -orbit containing infinitely many distinct $m$ -th powers. For $m\geqslant 5$ we show that the only such functions are those of the form $cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$ with $\unicode[STIX]{x1D713}\in K(x)$ , and for $m\leqslant 4$ we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.

With additional analysis, we show that the index set $\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$ is a union of finitely many arithmetic progressions, where $\unicode[STIX]{x1D719}^{n}$ denotes the $n$ -th iterate of $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D706}\in K(x)$ is any map Möbius-conjugate over $K$ to $x^{m}$ . When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^{m}=\unicode[STIX]{x1D719}^{n}(x)$ . We describe all $\unicode[STIX]{x1D719}$ for which these curves have an irreducible component of genus at most 1, and show that such $\unicode[STIX]{x1D719}$ must have two distinct iterates that are equal in $K(x)^{\ast }/K(x)^{\ast m}$ .

Hide All

Authors J. C. and J. S. research was supported by Carleton College’s HHMI grant for undergraduate science education and the Carleton College department of Mathematics and Statistics.

Hide All
[1] An, T. T. H. and Diep, N. T. N., Genus one factors of curves defined by separated variable polynomials . J. Number Theory 133(2013), no. 8, 26162634.
[2] Avanzi, R. M. and Zannier, U. M., Genus one curves defined by separated variable polynomials and a polynomial Pell equation . Acta Arith. 99(2001), 227256.
[3] Avanzi, R. M. and Zannier, U. M., The equation f (X) = f (Y) in rational functions X = X (t), Y= Y (t) . Compositio Math. 139(2003), no. 3, 263295.
[4] Beardon, A. F., Iteration of rational functions. Complex analytic dynamical systems. Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.
[5] Bell, J. P., Ghioca, D., and Tucker, T. J., The dynamical Mordell-Lang conjecture. Mathematical Surveys and Monographs, 210, American Mathematical Society, Providence, RI, 2016.
[6] Bilu, Y. F. and Tichy, R. F., The Diophantine equation f (x) = g (y) . Acta Arith. 95(2000), no. 3, 261288.
[7] Cremona, J., The elliptic curve database for conductors to 130000 . In: Algorithmic number theory, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006, pp. 1129.
[8] Fried, M. D., Arithmetical properties of function fields. II. The generalized Schur problem . Acta Arith. 25(1973/74), 225258.
[9] Ghioca, D., Tucker, T. J., and Zieve, M. E., Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture . Invent. Math. 171(2008), 463483.
[10] Ghioca, D., Tucker, T. J., and Zieve, M. E., Linear relations between polynomial orbits . Duke Math. J. 161(2012), no. 7, 13791410.
[11] Gratton, C., Nguyen, K., and Tucker, T. J., ABC implies primitive prime divisors in arithmetic dynamics . Bull. Lond. Math. Soc. 45(2013), 11941208.
[12] Hindry, M. and Silverman, J. H., Diophantine geometry: an introduction. Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000.
[13] Karpilovsky, G., Topics in field theory. North-Holland Mathematics Studies, 155, Notas de Matemática [Mathematical Notes], 124, North-Holland Publishing Co., Amsterdam, 1989.
[14] Lang, S., Number theory. III. Diophantine geometry. Encyclopaedia of Mathematical Sciences, 60, Springer-Verlag, Berlin, 1991.
[15] Milnor, J., On Lattès maps . In: Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 943.
[16] Northcott, D. G., Periodic points on an algebraic variety . Ann. of Math. (2) 51(1950), 167177.
[17] Pakovich, F., Algebraic curves A °l (x) - U (y) = 0 and arithmetic of orbits of rational functions. 2018. arxiv:1801.01985.
[18] Pakovich, F., Algebraic curves P (x) - Q (y) = 0 and functional equations . Complex Var. Elliptic Equ. 56(2011), no. 1–4, 199213.
[19] Ritt, J. F., Prime and composite polynomials . Trans. Amer. Math. Soc. 23(1922), no. 1, 5166.
[20] Schinzel, A., Selected topics on polynomials. University of Michigan Press, Ann Arbor, Mich., 1982.
[21] Silverman, J. H., Integer points, Diophantine approximation, and iteration of rational maps . Duke Math. J. 71(1993), no. 3, 793829.
[22] Silverman, J. H., The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241, Springer, New York, 2007.
[23] Stichtenoth, H., Algebraic function fields and codes. Second ed., Graduate Texts in Mathematics, 254, Springer-Verlag, Berlin, 2009.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed