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# Mathematical Research Letters

## Volume 16 (2009)

### Number 1

### Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

Pages: 87 – 101

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n1.a9

#### Authors

#### Abstract

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 pairs $(y_0,c)\in\CC^2$ for which $[K(y_0,c)\col K]<2^{N-3}$ and the value of the $N\tth$ iterate of $f_c(x)$ at $x=y_0$ is $a$. Moreover, the bound $2^{N-3}$ in this result is optimal.