## Involve: A Journal of Mathematics

• Involve
• Volume 5, Number 4 (2012), 481-495.

### Newly reducible iterates in families of quadratic polynomials

#### Abstract

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

$g γ , m ( x ) = ( x − γ ) 2 + m + γ , m ∈ K .$

For fixed $n≥3$ and nearly all values of $γ$, we show that there are only finitely many $m$ such that $gγ,m$ has a newly reducible $n$-th iterate. For $n=2$ we show a similar result for a much more restricted set of $γ$. 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.

#### Article information

Source
Involve, Volume 5, Number 4 (2012), 481-495.

Dates
Revised: 19 February 2013
Accepted: 4 April 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733539

Digital Object Identifier
doi:10.2140/involve.2012.5.481

Mathematical Reviews number (MathSciNet)
MR3069050

Zentralblatt MATH identifier
1348.11082

#### Citation

Chamberlin, Katharine; Colbert, Emma; Frechette, Sharon; Hefferman, Patrick; Jones, Rafe; Orchard, Sarah. Newly reducible iterates in families of quadratic polynomials. Involve 5 (2012), no. 4, 481--495. doi:10.2140/involve.2012.5.481. https://projecteuclid.org/euclid.involve/1513733539

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