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(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

gγ,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.

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