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The integer transfinite diameter of intervals and totally real algebraic integers. (English) Zbl 0892.11033
Given an interval \(I\) of the real line, the integer transfinite diameter of \(I\) is defined by \(t_{\mathbb Z}(I) = \inf_P \| P\| _\infty^{1/\partial P}\), where \(\| P\| _\infty\) is the maximum of \(| P(x)| \) on \(I\) and the infimum is over all non-constant \(P\) with integer coefficients. As usual \(\partial P\) denotes the degree of \(P\). The classical transfinite diameter \(t(I)\) has the same definition except that the infimum is over all polynomials with complex coefficients. It is known that \(t(I) = | I| /4\) and for intervals of length greater than \(4\) that \(t_{\mathbb Z}(I) = t(I)\). On the other hand the exact value of \(t_{\mathbb Z}(I)\) is not known for any interval of length less than \(4\).
This interesting paper studies \(t_{\mathbb Z}(I)\) for small intervals with one fixed rational endpoint. The authors define and study certain functions \(g_-,g, g_+\) which give lower and upper bounds for \(t_{\mathbb Z}(I)\) for Farey intervals \([p/q,r/s]\) with \(qr-ps = 1\). These are used to prove good upper and lower bounds for \(t_{\mathbb Z}(I)\) for intervals of the form \([r/s,r/s + \delta]\) or \([r/s - \delta,r/s]\), where \(r/s\) is fixed and \(\delta \to 0\). These results generalize those of F. Amoroso [Ann. Inst. Fourier 40, 885–911 (1990; Zbl 0713.41004)] and P. Borwein and T. Erdélyi [Math. Comput. 68, 661–681 (1996; Zbl 0859.11044)].
Extending results of the latter authors, they define critical polynomials \(P\) and critical values \(c_P(I)\) for an interval \(I\) which have the property that if \(Q \in \mathbb Z[x]\) has \(\| Q\| _\infty^{1/\partial Q} < c_P\) then \(P^k\) divides \(Q\), where \(k \geq \gamma \partial Q\), \(\gamma\) being a constant depending only on \(P\) and \(\| Q\| _\infty\). They determine 10 critical polynomials for \([0,1]\), extending results of E. Aparicio [J. Approximation Theory 55, 270–278 (1988; Zbl 0663.41008)] and Borwein and Erdélyi (loc. cit.). Finally they obtain from their results an interesting result concerning totally real algebraic integers, namely that if such an \(\alpha\) has least conjugate \(\alpha_1\), then the mean value of \(\alpha\), \(\text{Trace} (\alpha)/\partial \alpha > 1.6 + \alpha_1\), with 8 explicitly listed exceptions.

MSC:
11R04 Algebraic numbers; rings of algebraic integers
11R09 Polynomials (irreducibility, etc.)
11R80 Totally real fields
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