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Rational preimages in families of dynamical systems. (English) Zbl 1302.37060
Summary: Let $${\phi}$$ be a rational function of degree at least two defined over a number field $$k$$. Let $${a \in \mathbb{P}^1(k)}$$ and let $$K$$ be a number field containing $$k$$. We study the cardinality of the set of rational iterated preimages Preim$${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a \text{ for some } N \geq 1\}}$$. We prove two new results (Theorems 2 and 4) bounding $${|\mathrm {Preim}(\phi, a, K)|}$$ as $${\phi}$$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim$${(\phi, a, K)}$$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.

MSC:
 37P15 Dynamical systems over global ground fields 14G25 Global ground fields in algebraic geometry 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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References:
 [1] Baker M.: A finiteness theorem for canonical heights attached to rational maps over function fields. J. Reine Angew. Math. 626, 205–233 (2009) · Zbl 1187.37133 [2] Bilu Y., Parent P.: Runge’s method and modular curves. Int. Math. Res. Notices IMRN 2011(9), 1997–2027 (2011) · Zbl 1304.11054 [3] Bilu Y., Parent P.: Serre’s uniformity problem in the split Cartan case. Ann. Math. (2) 173(1), 569–584 (2011) · Zbl 1278.11065 · doi:10.4007/annals.2011.173.1.13 [4] Bombieri E.: On Weil’s ”théorème de décomposition”. Am. J. Math. 105(2), 295–308 (1983) · Zbl 0516.12009 · doi:10.2307/2374261 [5] Call G.S., Silverman J.H.: Canonical heights on varieties with morphisms. Composit. Math. 89(2), 163–205 (1993) · Zbl 0826.14015 [6] Evertse J.H.: The number of solutions of decomposable form equations. Invent. Math. 122(3), 559–601 (1995) · Zbl 0851.11019 · doi:10.1007/BF01231456 [7] Faber X.: A remark on the effective Mordell conjecture and rational pre-images under quadratic dynamical systems. C. R. Math. Acad. Sci. Paris 348(7–8), 355–358 (2010) · Zbl 1264.37053 · doi:10.1016/j.crma.2010.02.010 [8] Faber X., Hutz B., Ingram P., Jones R., Manes M., Tucker T.J., Zieve M.E.: Uniform bounds on pre-images under quadratic dynamical systems. Math. Res. Lett. 16(1), 87–101 (2009) · Zbl 1222.11086 · doi:10.4310/MRL.2009.v16.n1.a9 [9] Faber, X., Hutz, B., Stoll, M.: On the number of rational iterated pre-images of the origin under quadratic dynamical systems. Int. J. Number Theory (2012, to appear) · Zbl 1242.14019 [10] Fakhruddin N.: Questions on self maps of algebraic varieties. J. Ramanujan Math. Soc. 18(2), 109–122 (2003) · Zbl 1053.14025 [11] Hutz, B., Hyde, T., Krause, B.: Pre-images of quadratic dynamical systems. Involve (2012, to appear) · Zbl 1258.37076 [12] Ingram P.: Lower bounds on the canonical height associated to the morphism $${$$\backslash$$phi(z) = z\^d +c}$$ . Monatsh. Math. 157(1), 69–89 (2009) · Zbl 1239.11071 · doi:10.1007/s00605-008-0018-6 [13] Levin A.: Variations on a theme of Runge: effective determination of integral points on certain varieties. J. Théor. Nombres Bordeaux 20(2), 385–417 (2008) · Zbl 1179.11018 · doi:10.5802/jtnb.634 [14] Manin J.I.: The p-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR Ser. Mat. 33, 459–465 (1969) · Zbl 0191.19601 [15] Merel L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124(1–3), 437–449 (1996) · Zbl 0936.11037 · doi:10.1007/s002220050059 [16] Morton P., Silverman J.H.: Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math. 461, 81–122 (1995) · Zbl 0813.11059 [17] Runge C.: Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math. 100, 425–435 (1887) · JFM 19.0076.03 [18] Silverman J.H.: The space of rational maps on $${$$\backslash$$mathbb{P}\^1}$$ . Duke Math. J. 94(1), 41–77 (1998) · Zbl 0966.14031 · doi:10.1215/S0012-7094-98-09404-2 [19] Silverman J.H.: The arithmetic of dynamical systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007) · Zbl 1130.37001 [20] Vojta P.: Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, vol. 1239. Springer, Berlin (1987) · Zbl 0609.14011
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