Height lower bounds for elements in highly composite rings

Abstract: Let $\alpha$ be a non-zero algebraic number and not a root of unity. Then its absolute Weil height $h(\alpha)$ is positive. How small can it be in terms of the degree of $\alpha$? Lehmer's conjecture states that the height does not decay faster than the reciprocal of the degree. Amoroso and Dvornicich showed that there is no decay at all, provided $\mathbf{Q}(\alpha)/\mathbf{Q}$ is an abelian extension. If we further require that $\alpha$ lies in $\mathbf{Q}^{(d)}$, the composite field of all number fields of degree at most $d$, then the height tends to infinity, as was shown by Bombieri and Zannier. What happens for arbitrary $\alpha \in \mathbf{Q}^{(d)}$? We will present some new results on this unsolved question. This is ongoing joint work with Siu Hang Man, Niclas Technau, and Pavlo Yatsyna.

Seminarraum 05.002, Spiegelgasse 5