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Bounded heights in families of dynamical systems Nguyen, Dang Khoa


Let $a,b\in\bar{\mathbb{Q}}$ be such that exactly one of $a$ and $b$ is an algebraic integer, and let$f_t(z)=z^2+t$be a family of quadratic polynomials parametrized by $t\in\bar{\mathbb{Q}}$. We prove that the set of all $t\in\bar{\mathbb{Q}}$ for which there exist $m,n\geq 0$ such that $f_t^m(a)=f_t^n(b)$ has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics. This is joint work with DeMarco, Ghioca, Krieger, Tucker, and Ye.

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