Algebra & Number Theory

A dynamical variant of the Pink–Zilber conjecture

Dragos Ghioca and Khoa Dang Nguyen

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

dynamical Pink–Zilber conjecture heights


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.

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  • 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,
  • 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.