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# The Infimum in the Metric Mahler Measure

Published:2011-03-05
Printed: Dec 2011
• Charles L. Samuels,
Simon Fraser University, Department of Mathematics, Burnaby, BC V5A1S6
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## Abstract

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.
 Keywords: Weil height, Mahler measure, metric Mahler measure, Lehmer's problem
 MSC Classifications: 11R04 - Algebraic numbers; rings of algebraic integers 11R09 - Polynomials (irreducibility, etc.)

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