# Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

@article{Faber2008UniformBO, title={Uniform Bounds on Pre-Images under Quadratic Dynamical Systems}, author={X. Faber and Benjamin Hutz and P. Ingram and R. Jones and M. Manes and T. Tucker and Michael E. Zieve}, journal={Mathematical Research Letters}, year={2008}, volume={16}, pages={87-101} }

For any elements $a,c$ of a number field $K$, let $\Gamma(a,c)$ denote the backwards orbit of $a$ under the map $f_c\colon\CC\to\CC$ given by $f_c(x)=x^2+c$. We prove an upper bound on the number of elements of $\Gamma(a,c)$ whose degree over $K$ is at most some constant $B$. This bound depends only on $a$, $[K:\QQ]$, and $B$, and is valid for all $a$ outside an explicit finite set. We also show that, for any fixed $N\ge 4$ and any $a\in K$ outside a finite set, there are only finitely many… CONTINUE READING

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