#### Vol. 5, No. 4, 2012

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Newly reducible iterates in families of quadratic polynomials

### Katharine Chamberlin, Emma Colbert, Sharon Frechette, Patrick Hefferman, Rafe Jones and Sarah Orchard

Vol. 5 (2012), No. 4, 481–495
##### Abstract

We examine the question of when a quadratic polynomial $f\left(x\right)$ defined over a number field $K$ can have a newly reducible $n$-th iterate, that is, ${f}^{n}\left(x\right)$ irreducible over $K$ but ${f}^{n+1}\left(x\right)$ reducible over $K$, where ${f}^{n}$ denotes the $n$-th iterate of $f$. For each choice of critical point $\gamma$, we consider the family

${g}_{\gamma ,m}\left(x\right)={\left(x-\gamma \right)}^{2}+m+\gamma ,\phantom{\rule{1em}{0ex}}m\in K.$

For fixed $n\ge 3$ and nearly all values of $\gamma$, we show that there are only finitely many $m$ such that ${g}_{\gamma ,m}$ has a newly reducible $n$-th iterate. For $n=2$ we show a similar result for a much more restricted set of $\gamma$. These results complement those obtained by Danielson and Fein (Proc. Amer. Math. Soc. 130:6 (2002), 1589–1596) in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings’ theorem.

##### Keywords
polynomial iteration, polynomial irreducibility, arithmetic dynamics, rational points on hyperelliptic curves
##### Mathematical Subject Classification 2010
Primary: 11R09, 37P05, 37P15