Newly reducible iterates in families of quadratic polynomials

TL;DR

This paper investigates when quadratic polynomials over number fields have a newly reducible iterate, showing finiteness results for such cases using hyperelliptic curves and Faltings' theorem.

Contribution

It provides new finiteness results for newly reducible iterates of quadratic polynomials, extending previous work to specific families and iterates.

Findings

Finitely many parameters yield newly reducible nth iterates for fixed n ≥ 3.

Similar finiteness results hold for n=2 under restricted conditions.

Uses hyperelliptic curves and Faltings' theorem to establish results.

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 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.