Norms extremal with respect to the Mahler measure

TL;DR

This paper introduces new vector space norms related to the Mahler measure that satisfy extremal properties, generalizing previous metrics and providing explicit evaluations and finite-dimensional attainment results.

Contribution

It develops a family of norms extremal with respect to the Mahler measure, extending the metric Mahler measure and establishing finite-dimensional achievement.

Findings

Norms are constructed with extremal properties relative to Mahler measure.

Explicit evaluation of these norms on specific algebraic numbers.

Proof that the infimum defining the norms is attained in finite-dimensional space.

Abstract

In a previous paper, the authors introduced several vector space norms on the space of algebraic numbers modulo torsion which corresponded to the Mahler measure on a certain class of numbers and allowed the authors to formulate L^p Lehmer conjectures which were equivalent to their classical counterparts. In this paper, we introduce and study several analogous norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We evaluate these norms on certain classes of algebraic numbers and prove that the infimum in the construction is achieved in a certain finite dimensional space.