Abstract
Let M(α) be the Mahler measure of an algebraic number α and let G(α) be the modulus of the product of logarithms of absolute values of its conjugates. We prove that if α is a nonreciprocal algebraic number of degree d ≥ 2 then M(α)2 G(α)1/d ≥ 1/2d. This estimate is sharp up to a constant. As a main tool for the proof we develop an idea of Cassels on an estimate for the resultant of α and 1/α. We give a number of immediate corollaries, e.g., some versions of Smyth's inequality for the Mahler measure of a nonreciprocal algebraic integer from below.
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Dubickas, A. On the Measure of a Nonreciprocal Algebraic Number. The Ramanujan Journal 4, 291–298 (2000). https://doi.org/10.1023/A:1009801120348
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DOI: https://doi.org/10.1023/A:1009801120348