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A Collection of Metric Mahler Measures


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1 University of British Columbia, Department of Mathematics, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
     

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Let M(α) denote the Mahler measure of the algebraic number α. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version.We show how to generalize the above constructions in order to associate, to each point in (0,∞], a metric version Mx of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute Mx (α) for sufficiently small x, identifying, in the process, a function ¯M with certain minimality properties. Further, we show that the map xMx (α) defines a continuous function on the positive real numbers.
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  • A Collection of Metric Mahler Measures

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Authors

Charles L. Samuels
University of British Columbia, Department of Mathematics, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

Abstract


Let M(α) denote the Mahler measure of the algebraic number α. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version.We show how to generalize the above constructions in order to associate, to each point in (0,∞], a metric version Mx of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute Mx (α) for sufficiently small x, identifying, in the process, a function ¯M with certain minimality properties. Further, we show that the map xMx (α) defines a continuous function on the positive real numbers.