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Keywords:
Mahler measure; quadrinomials; irreducibility; nonreciprocal numbers
Mahler measure; quadrinomials; irreducibility; nonreciprocal numbers
Summary:
The main result of this paper implies that for every positive integer $d\geqslant 2$ there are at least $(d-3)^2/2$ nonconjugate algebraic numbers which have their Mahler measures lying in the interval $(1,2)$. These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.
References:
[1] Boyd D.W.: Perron units which are not Mahler measures. Ergodic Theory Dynam. Systems 6 (1986), 485-488. MR 0873427 | Zbl 0591.12003
[2] Boyd D.W., Montgomery H.L.: Cyclotomic Partitions. Number Theory (R.A. Mollin, ed.), Walter de Gruyter, Berlin, 1990, pp.7-25. MR 1106647 | Zbl 0697.10040
[3] Chern S.J., Vaaler J.D.: The distribution of values of Mahler's measure. J. Reine Angew. Math. 540 (2001), 1-47. MR 1868596 | Zbl 0986.11017
[4] Dixon J.D., Dubickas A.: The values of Mahler measures. Mathematika, to appear. MR 2220217 | Zbl 1107.11044
[5] Dubickas A.: Algebraic conjugates outside the unit circle. New Trends in Probability and Statistics Vol. 4: Analytic and Probabilistic Methods in Number Theory (A. Laurinčikas et al., eds.), Palanga, 1996, TEV Vilnius, VSP Utrecht, 1997, pp.11-21. MR 1653596 | Zbl 0923.11146
[6] Dubickas A., Konyagin S.V.: On the number of polynomials of bounded measure. Acta Arith. 86 (1998), 325-342. MR 1659085 | Zbl 0926.11080
[7] Ljunggren W.: On the irreducibility of certain trinomials and quadrinomials. Math. Scand. 8 (1960), 65-70. MR 0124313 | Zbl 0095.01305
[8] Mignotte M.: Sur les nombres algébriques de petite mesure. Comité des Travaux Historiques et Scientifiques: Bul. Sec. Sci. 3 (1981), 65-80. MR 0638732 | Zbl 0467.12008
[9] Mignotte M.: On algebraic integers of small measure. Topics in Classical Number Theory, Budapest, 1981, vol. II (G. Halász, ed.), Colloq. Math. Soc. János Bolyai 34 (1984), 1069-1077. MR 0781176 | Zbl 0543.12002
[10] Schinzel A.: Polynomials with Special Regard to Reducibility. Encyclopedia of Mathematics and its Applications 77, Cambridge University Press, 2000. MR 1770638 | Zbl 0956.12001
[11] Smyth C.J.: On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169-175. MR 0289451 | Zbl 0235.12003
[12] Smyth C.J.: Topics in the theory of numbers. Ph.D. Thesis, University of Cambridge, 1972.
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