Newly reducible iterates in families of quadratic polynomials

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

doi:10.2140/involve.2012.5.481

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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

DOI: 10.2140/involve.2012.5.481

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, $f^{n} (x)$ irreducible over $K$ but $f^{n + 1} (x)$ reducible over $K$ , where $f^{n}$ 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 \in K .

For fixed $n \geq 3$ and nearly all values of $γ$ , we show that there are only finitely many $m$ such that $g_{γ, m}$ 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.

Keywords

polynomial iteration, polynomial irreducibility, arithmetic dynamics, rational points on hyperelliptic curves

Mathematical Subject Classification 2010

Primary: 11R09, 37P05, 37P15

Milestones

Received: 15 October 2012

Revised: 19 February 2013

Accepted: 4 April 2013

Published: 14 June 2013

Communicated by Michael Zieve

Authors

Katharine Chamberlin
Department of Mathematics and Computer Science College of the Holy Cross One College Street Worcester, MA 10610 United States

Emma Colbert
Department of Mathematics and Computer Science College of the Holy Cross One College Street Worcester, MA 01610 United States

Sharon Frechette
Department of Mathematics and Computer Science College of the Holy Cross One College Street Worcester, MA 01610 United States

Patrick Hefferman
Department of Mathematics and Computer Science College of the Holy Cross One College Street Worcester, MA 01610 United States

Rafe Jones
Department of Mathematics Carleton College One North College Street Northfield, MN 55057 United States

Sarah Orchard
Department of Mathematics and Computer Science College of the Holy Cross One College Street Worcester, MA 01610 United States

Vol. 5, No. 4, 2012