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On intersections of polynomial semigroups orbits with plane lines

Part of: Polynomials and matrices Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  17 July 2020

Jorge Mello
Jorge Mello*
Affiliation:
University of New South Wales, School of Mathematics and Statistics, UNSW Sydney, NSW2052, Australia
*
e-mail: j.mello@unsw.edu.au
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Abstract

We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of D. Ghioca, T. Tucker, and M. Zieve (2008).

Keywords

Polynomial semigroup dynamicsunlikely intersections in arithmetic dynamics

MSC classification

Primary: 11C08: Polynomials 14G25: Global ground fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 442 - 451
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author’s work was supported by the ARC Discovery Grant DP180100201 and UNSW.

References

Bell, J. P., Ghioca, D., and Tucker, T. J., The dynamical Mordell-Lang problem for Noetherian spaces . Funct. Approx. Comment. Math. 53(2015), no. 2, 313–328. http://dx.doi.org/10.7169/facm/2015.53.2.7 CrossRefGoogle Scholar
Bell, J. P., Ghioca, D., and Tucker, T. J., The dynamical Mordell-Lang conjecture . Mathematical surveys and monographs, 210, Amer. Math. Soc., Providence, RI, 2016.Google Scholar
Benedetto, R., Ghioca, D., Kurlberg, P., and Tucker, T., A gap principle for dynamics . Compos. Math. 146(2010), no. 4, 1056–1072. http://dx.doi.org/10.1112/S0010437X09004667 CrossRefGoogle Scholar
Benedetto, R., Ghioca, D., Kurlberg, P., and Tucker, T., A case of the dynamical Mordell-Lang conjecture . Math. Ann. 352(2012), 1–26. http://dx.doi.org/10.1007/s00208-010-0621-4 CrossRefGoogle Scholar
Bilu, Y. and Tichy, R., The diophantine equation $f(x)=g(y)$ . Acta Arith. 95(2000), 261–288. http://dx.doi.org/10.4064/aa-95-3-261-288 CrossRefGoogle Scholar
D’Andrea, C., Ostafe, A., Shparlinski, I. E., and Sombra, M., Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems . Trans. Am. Math. Soc. 371(2019), 1169–1198. http://dx.doi.org/10.1090/tran/7437 CrossRefGoogle Scholar
Ghioca, D. and Nguyen, K. D., The orbit intersection problem for linear spaces and semiabelian varieties . Math. Res. Lett. 24(2017), no. 5, 1263–1283. http://dx.doi.org/10.4310/MRL.2017.v24.n5.a2 CrossRefGoogle Scholar
Ghioca, D., Tucker, T., and Zieve, M., Intersections of polynomial orbits, and a dynamical Mordell–Lang conjecture . Invent. Math. 171(2008), 463–483. http://dx.doi.org/10.1007/s00222-007-0087-5 CrossRefGoogle Scholar
Ghioca, D., Tucker, T., and Zieve, M., The Mordell-Lang question for endomorphisms of semiabelian varieties . J. Théor. Nombres Bordeaux 23(2011), no. 3, 645–666. http://dx.doi.org/10.5802/jtnb.781 CrossRefGoogle Scholar
Ghioca, D., Tucker, T., and Zieve, M., Linear relations between polynomial orbits . Duke Math. J. 161(2012), 1379–1410. http://dx.doi.org/10.1215/00127094-1598098 CrossRefGoogle Scholar
Kawaguchi, S., Canonical heights, invariant currents, and dynamical systems of morphisms associated with line bundles . J. Reine Angew. Math. 597(2006), 135–173. http://dx.doi.org/10.1515/CRELLE.2006.065 Google Scholar
Kawaguchi, S., Canonical heights for random iterations in certain varieties . Int. Math. Res. Not. (2007), rnm 023. http://dx.doi.org/10.1093/imrn/rnm023 Google Scholar
Ostafe, A. and Sha, M., On the quantitative dynamical Mordell–Lang conjecture . J. Number Theory 156(2015), 161–182. http://dx.doi.org/10.1016/j.jnt.2015.04.011 CrossRefGoogle Scholar
Ostafe, A. and Shparlinski, I. E., Orbits of algebraic dynamical systems in subgroups and subfields . In: Number theory—Diophantine problems, uniform distribution and applications, Springer, Cham, 2017, pp. 347–368.CrossRefGoogle Scholar
Wang, M. X., A dynamical Mordell-Lang property on the disk . Trans. Am. Math. Soc. 369(2017), no. 3, 2183–2204. http://dx.doi.org/10.1090/tran/6775 CrossRefGoogle Scholar