# Mahler measures of values of polynomials

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 $$M(f(\alpha))$$?

• 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 $M(\alpha)$. – Gerry Myerson Apr 15 at 5:27
• In which sense do you want to compare the values? Do you mean inequalities or something else? – François Brunault Apr 20 at 8:51
• @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. – Maurizio Moreschi Apr 20 at 12:43