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Abstract

Let α be a number algebraic over the rationals and let H(α) denote the absolute logarithmic height of α, which can be defined as H(α)= \\log M(f)\\sp1/n, where α is a root of the irreducible polynomial f(x) with rational coefficients and degree n, and where M(f) denotes the Mahler measure of f. The author gives an elementary proof of a sharp version of a remarkable inequality of Shouwu Zhang. He shows that, for all α\\ne 0,1, (1\\pm\\sqrt-3)/2, the following inequality holds: H(α)+ H(1-α)≥q 1\\over 2 \\log 1+\\sqrt5\\over 2= 0.2406059\\dots, with equality if and only if α or 1-α is a primitive 10th root of unity. He also proves a sharp projective version of this inequality for the curve x+y+z=0 and gives an outline of how to prove similar results for other curves.