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 nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f. For each choice of critical point \gamma of f(x), we consider the family g_{\gamma,m}(x)= (x  \gamma)^2 + m + \gamma, m \in K. For fixed n \geq 3 and nearly all values of \gamma, we show that there are only finitely many m such that g_{\gamma,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of \gamma. These results complement those obtained by Danielson and Fein in the higherdegree 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.4127
 Bibcode:
 2012arXiv1210.4127C
 Keywords:

 Mathematics  Number Theory;
 37P05;
 11R09;
 37P15
 EPrint:
 13 pages, one figure, one appendix