A dynamical analogue of a question of Fermat
DOI:
https://doi.org/10.7146/math.scand.a-142342Abstract
Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.
References
Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. https://doi.org/10.1006/jsco.1996.0125
Barsakçi, B., and Sadek, M., Simultaneous rational periodic points of degree-$2$ rational maps, J. Number Theory 243 (2023), 715–728. https://doi.org/10.1016/j.jnt.2022.06.005
Cahn, J., Jones, R., and Spear, J., Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture, Canad. J. Math. 71 (2019), no. 4, 773–817. https://doi.org/10.4153/cjm-2018-026-x
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366. https://doi.org/10.1007/BF01388432
Flynn, E. V., Poonen, B., Schaefer, E. F., Cycles of quadratic polynomials and rational points on a genus-$2$ curve, Duke Math. J. 90 (1997), no. 3, 435–463. https://doi.org/10.1215/S0012-7094-97-09011-6
González-Jiménez, E., and Xarles, X. Five squares in arithmetic progression over quadratic fields, Rev. Mat. Iberoam. 29 (2013), no. 4, 1211–1238. https://doi.org/10.4171/RMI/754
Hindes, W., Finite orbit points for sets of quadratic polynomials, Int. J. Number Theory 15 (2019), no. 8, 1693–1719. https://doi.org/10.1142/S1793042119500945
Ingram, P., Canonical heights and preperiodic points for certain weighted homogeneous families of polynomials, Int. Math. Res. Not. IMRN 2019, no. 15, 4859–4879. https://doi.org/10.1093/imrn/rnx291
Morton, P., Arithmetic properties of periodic points of quadratic maps, II, Acta Arith. 87 (1998), no. 2, 89–102. https://doi.org/10.4064/aa-87-2-89-102
Poonen, B., The classification of rational preperiodic points of quadratic polynomials over $mathbb Q$: a refined conjecture, Math. Z. 228 (1998), no. 1, 11–29. https://doi.org/10.1007/PL00004405
Sadek, M., Families of polynomials of every degree with no rational preperiodic points, C. R. Math. Acad. Sci. Paris 359 (2021), 195–197. https://doi.org/10.5802/crmath.173
Sadek, M., and Uyar, M., Boundedness results for periodic points of rational maps, preprint.
Stoll, M., Rational $6$-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380. https://doi.org/10.1112/S1461157000000644
Walde, W., and Russo, P., Rational periodic points of the quadratic function $mathbb Q_c(x) = x^2 + c$, Amer. Math. Monthly 101 (1994), no. 4, 318–331. https://doi.org/10.2307/2975624
Xarles, X., Squares in arithmetic progression over number fields, J. Number Theory 132 (2012), no. 3, 379–389. https://doi.org/10.1016/j.jnt.2011.07.010
