Dresden, Gregory P. Orbits of algebraic numbers with low heights. (English) Zbl 0926.11078 Math. Comput. 67, No. 222, 815-820 (1998). The author proves that if \(\alpha\) is an algebraic integer, then the smallest value of \(h(\alpha)+ h(\frac{1}{1-\alpha})+ h(1-\frac{1}{\alpha})\) are 0 and \(0.4218\dots\), where \(h(\alpha)\) is the logarithmic Weil height defined by \(h(\alpha)= \sum_\nu \log\max (|\alpha |_\nu,1)\), where \(| \alpha|_\nu\) is the absolute value associated with the completion \(K_\nu\) of an algebraic field \(K\). Reviewer: R.Mollin (Calgary) Cited in 1 ReviewCited in 4 Documents MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) Keywords:root of unity; algebraic integer; logarithmic Weil height PDFBibTeX XMLCite \textit{G. P. Dresden}, Math. Comput. 67, No. 222, 815--820 (1998; Zbl 0926.11078) Full Text: DOI References: [1] E. Bombieri, A van der Poorten, and J. Vaaler, Effective measures of irrationality for cubic extensions of number fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 211-248. CMP 97:08 · Zbl 0879.11035 [2] David W. Boyd, Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), no. 4, 453 – 469. · Zbl 0474.12005 · doi:10.4153/CMB-1981-069-5 [3] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391 – 401. · Zbl 0416.12001 [4] D. H. Lehmer, Factorization of certain cyclotomic funcions, Ann. of Math. 34 (1933), 461-479. · Zbl 0007.19904 [5] Christopher G. Pinner and Jeffrey D. Vaaler, The number of irreducible factors of a polynomial. I, Trans. Amer. Math. Soc. 339 (1993), no. 2, 809 – 834. · Zbl 0787.11045 [6] Ulrich Rausch, On a theorem of Dobrowolski about the product of conjugate numbers, Colloq. Math. 50 (1985), no. 1, 137 – 142. · Zbl 0579.12001 [7] D. Zagier, Algebraic numbers close to both 0 and 1, Math. Comp. 61 (1993), no. 203, 485 – 491. · Zbl 0786.11063 [8] Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569 – 587. · Zbl 0788.14017 · doi:10.2307/2946601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.