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Linear relations between polynomial orbits. (English) Zbl 1267.37043

This work concerns the orbits of two complex polynomials of degree greater than 1. The authors prove that if two such polynomials have orbits with infinite intersection, then they must have a common iterate, generalising an existing result for polynomials with the same degree. The authors also prove a more general theorem concerning the intersection of any line in \(\mathbb{C}^d\) with a \(d\)-uple of orbits of nonlinear polynomials. Finally, a question is formulated, which brings together a generalisation of the present work and the Mordell-Lang conjecture.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
11C08 Polynomials in number theory
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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References:

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