Uniform Bounds on Pre-images under Quadratic Dynamical Systems

@inproceedings{Faber2008UniformBO,
  title={Uniform Bounds on Pre-images under Quadratic Dynamical Systems},
  author={X. W. C. Faber and Benjamin Hutz and Patrick Ingram and Rafe Jones and Michelle Manes and Thomas J. Tucker and Michael E. Zieve},
  year={2008}
}
For any elements a, c of a number field K, let Γ(a, c) denote the backwards orbit of a under the map fc : C→ C given by fc(x) = x2 + c. We prove an upper bound on the number of elements of Γ(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also show that, for any fixed N ≥ 4 and any a ∈ K outside a finite set, there are only finitely many pairs (y0, c) ∈ C2 for which [K(y0, c) : K] < 2N… CONTINUE READING

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