Let $\alpha$ be a zero of a polynomial of degree $n$ with odd coefficients, with $\alpha$ not a root of unity. We show that the height of $\alpha$ satisfies$$h\left(\alpha \right)\ge \frac{0.4278}{n+1}.$$More generally, we obtain bounds when the coefficients are all congruent to $1$ modulo $m$ for some $m\ge 2$.

We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.