On sets of algebraic integers whose remaining conjugates lie in the unit circle. (English) Zbl 0135.09002

Let \((\theta_1,\theta_2,\cdots,\theta_k)\) be a \(k\)-tuple of distinct algebraic integers, each with absolute value greater than one. Let \(P(z)\) be the polynomial of least degree with relatively prime integer coefficients having \(\theta_1,\theta_2,\cdots,\theta_k\) as roots. If the remaining roots of \(P(z)\) lie in the open (closed) unit circle, then \((\theta_1,\theta_2,\cdots,\theta_k)\) is a PV \((T)\ (k)\)-tuple. The PV 1-tuples are the well-known PV (Pisot-Vijayaraghavan) numbers. In this paper, the author generalizes results established for PV \((T)\) 1-tuples and 2-tuples by [C. Pisot, Ann. Sc. Norm. Super. Pisa, II. Ser. 7, 205–248 (1938; Zbl 0019.15502; JFM 64.0994.01)], J. Dufresnoy and Ch. Pisot [Ann. Sci. Éc. Norm. Supér. (3) 70, 105–133 (1953; Zbl 0051.02904)], R. Salem [Duke Math. J. 11, 103–108 (1944; Zbl 0063.06657); ibid. 12, 153–172 (1945; Zbl 0060.21601)], P. A. Samet [Proc. Camb. Philos. Soc. 49, 421–436 (1953; Zbl 0050.26501)], and the reviewer [Am. J. Math. 72, 565–572 (1950; Zbl 0041.17704)]. Of particular interest is the following theorem, which extends the celebrated theorem of Salem stating that the set of PV numbers is closed: Let a convergent sequence of PV \(k\)-tuples have limit \((\theta_1,\theta_2,\cdots,\theta_k)\), with \(|\theta_i|\neq 1\), for \(1\leq i\leq k\). Suppose that \(k'\leq k\) of the numbers \(\theta_1,\theta_2,\cdots,\theta_k\) are distinct. Then some non-void subset of \((\theta_1,\theta_2,\cdots,\theta_k)\) containing \(k''\) elements is a PV \(k''\)-tuple, with \(1\leq k''\leq k'\). It is conjectured that one may take \(k''=k'\).


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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