Algebra & Number Theory

A dynamical variant of the Pink–Zilber conjecture

Dragos Ghioca and Khoa Dang Nguyen

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let f1,,fn¯[x] be polynomials of degree d>1 such that no fi is conjugate to xd or to ±Cd(x), where Cd(x) is the Chebyshev polynomial of degree d. We let φ be their coordinatewise action on An, i.e., φ:AnAn is given by (x1,,xn)(f1(x1),,fn(xn)). We prove a dynamical version of the Pink–Zilber conjecture for subvarieties V of An with respect to the dynamical system (An,φ), if min{dim(V),codim(V)1}1.

Article information

Source
Algebra Number Theory, Volume 12, Number 7 (2018), 1749-1771.

Dates
Received: 1 November 2017
Revised: 6 April 2018
Accepted: 6 June 2018
First available in Project Euclid: 9 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1541732440

Digital Object Identifier
doi:10.2140/ant.2018.12.1749

Mathematical Reviews number (MathSciNet)
MR3871509

Zentralblatt MATH identifier
06976301

Subjects
Primary: 11G50: Heights [See also 14G40, 37P30]
Secondary: 11G35: Varieties over global fields [See also 14G25] 14G25: Global ground fields

Keywords
dynamical Pink–Zilber conjecture heights

Citation

Ghioca, Dragos; Nguyen, Khoa Dang. A dynamical variant of the Pink–Zilber conjecture. Algebra Number Theory 12 (2018), no. 7, 1749--1771. doi:10.2140/ant.2018.12.1749. https://projecteuclid.org/euclid.ant/1541732440


Export citation

References

  • J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell–Lang conjecture, Mathematical Surveys and Monographs 210, American Mathematical Society, Providence, RI, 2016.
  • E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs 4, Cambridge University Press, 2006.
  • E. Bombieri, D. Masser, and U. Zannier, “Intersecting a curve with algebraic subgroups of multiplicative groups”, Internat. Math. Res. Notices 20 (1999), 1119–1140.
  • E. Bombieri, D. Masser, and U. Zannier, “Intersecting curves and algebraic subgroups: conjectures and more results”, Trans. Amer. Math. Soc. 358:5 (2006), 2247–2257.
  • E. Bombieri, D. Masser, and U. Zannier, “Anomalous subvarieties–-structure theorems and applications”, Int. Math. Res. Not. 2007:19 (2007), Art. ID rnm057, 33.
  • L. DeMarco, D. Ghioca, H. Krieger, K. D. Nguyen, T. J. Tucker, and H. Ye, “Bounded height in families of dynamical systems”, preprint, 2017. To appear in Int. Math. Res. Not.
  • D. Ghioca and K. D. Nguyen, “Dynamical anomalous subvarieties: structure and bounded height theorems”, Adv. Math. 288 (2016), 1433–1462.
  • D. Ghioca and K. D. Nguyen, “Dynamics of split polynomial maps: uniform bounds for periods and applications”, Int. Math. Res. Not. 2017:1 (2017), 213–231.
  • D. Ghioca, T. J. Tucker, and S. Zhang, “Towards a dynamical Manin–Mumford conjecture”, Int. Math. Res. Not. 2011:22 (2011), 5109–5122.
  • D. Ghioca, K. D. Nguyen, and H. Ye, “The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps”, preprint, 2015. To appear in J. Eur. Math. Soc. $($JEMS$)$.
  • D. Ghioca, K. D. Nguyen, and H. Ye, “The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of $\smash{(\mathbb P^1)^n}$”, Compos. Math. 154:7 (2018), 1441–1472.
  • A. Medvedev and T. Scanlon, “Invariant varieties for polynomial dynamical systems”, Ann. of Math. $(2)$ 179:1 (2014), 81–177.
  • K. Nguyen, “Some arithmetic dynamics of diagonally split polynomial maps”, Int. Math. Res. Not. 2015:5 (2015), 1159–1199.
  • R. Pink, “A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell-Lang”, preprint, https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf.
  • J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, Springer, 2007.
  • J. Xie, “The dynamical Mordell–Lang conjecture for polynomial endomorphisms of the affine plane”, pp. vi+110 in Journées de Géométrie Algébrique d'Orsay, Astérisque 394, Société Mathématique de France, Paris, 2017.
  • U. Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies 181, Princeton University Press, 2012.
  • S.-W. Zhang, “Distributions in algebraic dynamics”, pp. 381–430 in Surveys in differential geometry, Vol. X, edited by S. T. Yau, Surv. Differ. Geom. 10, International Press, Somerville, MA, 2006.
  • B. Zilber, “Exponential sums equations and the Schanuel conjecture”, J. London Math. Soc. $(2)$ 65:1 (2002), 27–44.