Preperiodic points for quadratic polynomials with small cycles over quadratic fields

TL;DR

This paper investigates the structure of preperiodic points for quadratic polynomials over quadratic fields, aiming to classify possible dynamical graphs and extend known results from rational to quadratic extensions.

Contribution

It advances the classification of preperiodic point graphs for quadratic polynomials over quadratic fields, building on prior work and providing partial results towards a broader dynamical uniform boundedness conjecture.

Findings

Partial classification of preperiodic graphs over quadratic fields

Extension of Poonen's classification from rationals to quadratic fields

New results constraining large-period points in quadratic extensions

Abstract

Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points of $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha) = \beta$. The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers $n \ge 1$ and $d \ge 2$, there are only finitely many isomorphism classes of directed graphs $G(f,K)$ as one ranges over all number fields $K$ of degree $n$ and polynomials $f(z) \in K[z]$ of degree $d$. In the case $(n,d) = (1,2)$, Poonen has given a complete classification of all directed graphs which may be realized as $G(f,\mathbb{Q})$ for some quadratic polynomial $f(z) \in \mathbb{Q}[z]$, under the assumption that $f$ does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case $(n,d) = (2,2)$. By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen's -- with a similar assumption on points of large period -- but over all quadratic extensions of $\mathbb{Q}$.