Linear relations between polynomial orbits

TL;DR

This paper investigates the intersections of polynomial orbits in complex space, proving that infinite intersections imply shared iterates, and extends the analysis to lines intersecting multiple orbits, connecting to broader conjectures.

Contribution

It establishes a new criterion linking infinite orbit intersections to common iterates for nonlinear polynomials and generalizes the problem to higher dimensions.

Findings

Infinite intersection of orbits implies common iterate for nonlinear polynomials.

Characterization of line intersections with multiple polynomial orbits in complex space.

Formulation of a generalized question relating to the Mordell--Lang conjecture.

Abstract

We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture.