Resumen de Uniform bounds on pre-images under quadratic dynamical systems
Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J. Tucker, Michael E. Zieve
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.
