Involve: A Journal of Mathematics

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

Newly reducible iterates in families of quadratic polynomials

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

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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 n3 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
Received: 15 October 2012
Revised: 19 February 2013
Accepted: 4 April 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 11R09: Polynomials (irreducibility, etc.) 37P05: Polynomial and rational maps 37P15: Global ground fields

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

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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