Vol. 12, No. 7, 2018

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A dynamical variant of the Pink–Zilber conjecture

Dragos Ghioca and Khoa Dang Nguyen

Vol. 12 (2018), No. 7, 1749–1771

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 : An An 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.

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dynamical Pink–Zilber conjecture, heights
Mathematical Subject Classification 2010
Primary: 11G50
Secondary: 11G35, 14G25
Received: 1 November 2017
Revised: 6 April 2018
Accepted: 6 June 2018
Published: 27 October 2018
Dragos Ghioca
Department of Mathematics
University of British Columbia
Vancouver BC
Khoa Dang Nguyen
Department of Mathematics and Statistics
The University of Calgary
Calgary AB