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

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 $10^9$.