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 Weil height, Mahler measure, metric Mahler measure, Lehmer's problem

MSC Classifications:

11R04, 11R09 show english descriptions Algebraic numbers; rings of algebraic integers
Polynomials (irreducibility, etc.) 11R04 - Algebraic numbers; rings of algebraic integers
11R09 - Polynomials (irreducibility, etc.)

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