On algebraic numbers of small measure
For a nonzero algebraic number \(\alpha\) of degree \(d\) and conjugates \(\alpha_1 = \alpha\), \(\alpha_2, \dots, \alpha_d\), denote by \(\house\alpha\) the maximum of the \(\alpha_i\)'s and by \(M(\alpha)\) its Mahler measure, i.e. the product \(q \prod^d_{i=1} \max \{1,|\alpha_i |\}\). It is shown that if \(\alpha\) is not a root of unity then for sufficiently large \(d\) one has \[ |\alpha-1 |> \exp \biggl(-(\pi/4 + \varepsilon) \bigl(d \log d \log M (\alpha) \bigr)^{1/2} \biggr) \] which improves a result of \textit{Y. Bugeaud}, \textit{M. Mignotte} and \textit{F. Normandin} [C. R. Acad. Sci., Paris, Sér. I 321, No. 5, 517-522 (1995; Zbl 0848.11047)] who had \((2/3)^{1/2}\) in place of \(\pi/4\). Moreover an upper bound for the number of real zeros of an irreducible polynomial over \(\mathbb{Z}\) is obtained in terms of the Mahler measure of its root and this is used to obtain for sufficiently large and all totally real \(\alpha\) of degree \(d\) and not of the form \(2\cos (\pi r)\) with rational \(r\) the inequality \[ \house\alpha > 2+3.8 {\log \log^3d \over d \log^4d}, \] which essentially improves a previous result of the author [Acta Arith. 63, 15-20 (1993; Zbl 0777.11039)].
upper bound for the number of real zeros of an irreducible polynomial
scientific article; zbMATH DE number 939538
scientific article