# Eventually stable quadratic polynomials over $\mathbb{Q}$

## TL;DR

This paper investigates the irreducibility patterns of iterates of quadratic polynomials over rationals, especially focusing on the family $f_{1/c}(x) = x^2 + 1/c$, and develops algorithms and methods to classify their stability properties.

## Contribution

The paper provides new criteria for irreducibility of iterates of $f_{1/c}$, algorithms for large bounds, and complete classifications for $|c| ext{ up to } 10^9$, extending understanding of polynomial stability.

## Key findings

- Identified all $c$ where the third iterate has at least four factors.
- Determined all $c$ with irreducible $f_{1/c}$ and third iterate with at least three factors.
- Developed polynomial-time algorithms for checking irreducibility conditions.

## Abstract

We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$ \textit{eventually stable} (over $\mathbb{Q}$). Previous work of Hamblen, Jones, and Madhu shows that $f_r$ is eventually stable unless $r$ has the form $1/c$ for some integer $c \not\in \{0,-1\}$, in which case existing methods break down. We study this family, and prove that several conditions on $c$ of various flavors imply that all iterates of $f_{1/c}$ are irreducible. We give an algorithm that checks the latter property for all $c$ up to a large bound $B$ in time polynomial in $\log B$. We find all $c$-values for which the third iterate of $f_{1/c}$ has at least four irreducible factors, and all $c$-values such that $f_{1/c}$ is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of $f_{1/c}$, for all $c$ with absolute value at most $10^9$.