# 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}
}
• X. Faber, +4 authors Michael E. Zieve
• Published 2008
• Mathematics
• Mathematical Research Letters
• 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