Abstract: Let $h$ be the (absolute logarithmic) Weil height. In 2000, Amoroso and Dvornicich proved that there is a positive constant $c > 0$ such that for any cyclotomic field $K$ and any $\alpha \in K$ with $h (\alpha ) \neq 0$, $h(\alpha) \geq c$. We can conjecture a dynamical version of this result as follows. Let $f$ be a polynomial of degree $d \geq 2$ defined over a number field $K$, and let $h_f$ be the associated canonical height for $f$. Then there exists a positive constant $c = c(f) > 0$ such that for any field $L / K$ generated by preperiodic points of $f$ and for any $\alpha \in L$ which is not preperiodic, $h_f (\alpha) \geq c$. I will discuss a possible approach for proving this conjecture. In particular, I will discuss my current progress in attempting to use this approach to find a new proof of Amoroso and Dvornicich's result.
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