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

@article{DeMark2019EventuallySQ, title={Eventually stable quadratic polynomials over \$\mathbb\{Q\}\$}, author={David DeMark and Wade Hindes and R. Jones and Moses Z. R. Misplon and M. Stoll and M. Stoneman}, journal={arXiv: Number Theory}, year={2019} }

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… CONTINUE READING

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