Involve: A Journal of Mathematics
- Volume 5, Number 4 (2012), 481-495.
Newly reducible iterates in families of quadratic polynomials
We examine the question of when a quadratic polynomial defined over a number field can have a newly reducible -th iterate, that is, irreducible over but reducible over , where denotes the -th iterate of . For each choice of critical point , we consider the family
For fixed and nearly all values of , we show that there are only finitely many such that has a newly reducible -th iterate. For we show a similar result for a much more restricted set of . 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.
Involve, Volume 5, Number 4 (2012), 481-495.
Received: 15 October 2012
Revised: 19 February 2013
Accepted: 4 April 2013
First available in Project Euclid: 20 December 2017
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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