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On diameters of algebraic integers. (English) Zbl 0693.12001

Let \(\alpha\) be an algebraic integer of degree \(n\ge 2\) with conjugates \(\alpha_1,\ldots,\alpha_n\). The diameter of \(\alpha\) is defined by \(\operatorname{diam}(\alpha)=\max\vert \alpha_i - \alpha_j\vert\). J. Favard, in 1928 [C. R. Acad. Sci. Paris 186, 1181–1182 (1928; JFM 54.0194.06)], proved that, if \(n=2\), then \(\operatorname{diam}(\alpha)\ge \sqrt{3}\), with equality for the cube root of \(1\) or \(-1\). He asked whether \(\operatorname{diam}(\alpha)\ge \sqrt{3}\) for all algebraic integers of degree \(\ge 2\).
In this direction, the authors prove the following two results:
Theorem 1: If \(\beta\) is a reciprocal algebraic integer of degree at least 2 and if \(\alpha =\beta +k\), for an integer \(k\), then \(\operatorname{diam}(\alpha)\ge \sqrt{3}\).
Theorem 2: If the degree of \(\alpha\) exceeds 200000, then \(\operatorname{diam}(\alpha)\ge 1.7321>\sqrt{3}\).
Although Theorem 2, in principle, reduces the question of Favard to a finite (but impractically large) amount of computation, this is now unnecessary, since M. Langevin, E. Reyssat and G. Rhin have proved that if \(n>2\), then \(\operatorname{diam}(\alpha)>\sqrt{3}\) [Ann. Inst. Fourier 38, No. 1, 1–16 (1988; Zbl 0634.12003)]. In addition M. Langevin has shown that \(\displaystyle\liminf_{n\to \infty}\operatorname{diam}(\alpha)=2\) [ibid. 38, No. 2, 1–10 (1988; Zbl 0634.12002)], which is the best possible result.
It should be observed that Theorem 1 was obtained in C.-W. Lloyd-Smith’s Ph. D. thesis [Problems on the distribution of conjugates of algebraic numbers, Univ. Adelaide (1980), cf. Bull. Aust. Math. Soc. 25, 303–304 (1982; Zbl 0474.12004)], and that Theorem 2 was announced in a 1982 seminar of P. E. Blanksby [Banach Cent. Publ. 17, 21–30 (1985; Zbl 0596.12019)]. See also the following review.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
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