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A generalization of a theorem of Schinzel. (English) Zbl 1084.11010
In 1973, Schinzel proved that the Mahler measure of a totally real algebraic integer \(\alpha \neq -1,0,1\) of degree \(d\) is at least \(((1+\sqrt {5})/2)^{d/2},\) whereas for totally positive algebraic integer \(\alpha \neq 1\) the lower bound is \(((1+\sqrt {5})/2)^d.\) Both these bounds are sharp. For instance, \(\beta=(3+\sqrt {5})/2\) is a totally positive algebraic integer of degree \(2,\) so \(M(\beta)=\beta=((1+\sqrt 5)/2)^2.\) There are several ways to generalize Schinzel’s result. One can either add an additional condition on \(\alpha\) and study the lower bound \(M(\alpha)^{1/d}\) for \(\alpha\) totally real and totally positive or search for a lower bound which involves not only the degree \(d\) of \(\alpha,\) but also some other parameters of \(\alpha,\) for example, its discriminant \(\Delta(\alpha).\) An example of such inequality was given by Zaimi in 1994. In this paper, the author proves a theorem of this kind. He shows that if \(\alpha\) is a totally positive algebraic integer of degree \(d \geq 2\) then \[ M(\alpha) \geq ((\delta + \sqrt {\delta^2+4})/2)^{d/2}, \] where \(\delta=| \Delta(\alpha)| ^{1/d(d-1)}.\) Note that the same example \(\beta=(3+\sqrt {5})/2\) with \(d=2\) and \(\delta=5^{1/2}\) shows that this inequality is sharp. In addition, the author proves a more technical theorem which involves not only the discriminant of \(\alpha,\) but also the constant coefficient of its minimal polynomial and compares these results with similar results obtained earlier. In the proof, one first finds the minimum of certain auxiliary function and then, since \(\alpha\) is an algebraic integer, combines this with relevant arithmetical information.
MSC:
11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)
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