Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}_K$ and let $f(X)\in \mathcal{O}_K[X]$. Denote the Mahler measure by $M$.

Is there any known result about the comparison of the values $M(\alpha)$ and $M(f(\alpha))$?

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    $\begingroup$ I don't know, but it would be good to get some trivial cases out of the way. E.g., if $f$ is a constant polynomial, then clearly $M(f(\alpha))$ has nothing to do with $M(\alpha)$. $\endgroup$ – Gerry Myerson Apr 15 at 5:27
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    $\begingroup$ In which sense do you want to compare the values? Do you mean inequalities or something else? $\endgroup$ – François Brunault Apr 20 at 8:51
  • $\begingroup$ @FrançoisBrunault Yes, I mean inequalities. For example, do constants $C, \, n$ (depending on $K$ and $f$, but not on $\alpha$) exist such that $M(f(\alpha))\le C M(\alpha)^n$? A bound in the other direction is probably way more problematic. $\endgroup$ – Maurizio Moreschi Apr 20 at 12:43

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