Abstract
For an algebraic number α, the metric Mahler measure \({m_1(\alpha)}\) was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure \({m_t(\alpha)}\) by the author [16]. The definition of \({m_t(\alpha)}\) involves taking an infimum over a certain collection N-tuples of points in \(\overline{\mathbb{Q}}\), and from previous work of Jankauskas and the author, the infimum in the definition of \({m_t(\alpha)}\) is attained by rational points when \({\alpha\in \mathbb{Q}}\). As a consequence of our main theorem in this article, we obtain an analog of this result when \({\mathbb{Q}}\) is replaced with any imaginary quadratic number field of class number equal to 1. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases.
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References
Borwein P., Dobrowolski E., Mossinghoff M. J.: Lehmer’s problem for polynomials with odd coefficients.. Ann. of Math., 166(2), 347–366 (2007)
H. Cohn, Advanced Number Theory, Dover Publications, Inc. (1980).
Dobrowolski E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith., 34, 391–401 (1979)
Dubickas A., Smyth C. J.: On the metric Mahler measure. J. Number Theory. 86, 368–387 (2001)
Dubickas A., Smyth C. J.: On metric heights. Period. Math. Hungar., 46(135–155), 46 135–155 (2003)
D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd edition, John Wiley and Sons, Inc. (2004).
Fili P., Samuels C. L.: On the non-Archimedean metric Mahler measure. J. Number Theory, 129, 1698–1708 (2009)
P. Furtwängler, Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper, Abh. Math. Sem. Hamburg, 7 (1930), 14–36.
Jankauskas J., Samuels C. L.: The t-metric Mahler measures of surds and rational numbers. Acta Math. Hungar., 134, 481–498 (2012)
F. Jarvis, Algebraic Number Theory, Undergraduate Mathematics Series, Springer (2014).
Lehmer D. H.: Factorization of certain cyclotomic functions. Ann. of Math., 34, 461–479 (1933)
J. S. Milne, Class Field Theory Course Notes, http://www.jmilne.org/math/
J. Neukirch, Algebraic Number Theory, Springer (1999).
Northcott D. G.: An inequality on the theory of arithmetic on algebraic varieties. Proc. Cambridge Philos. Soc., 45, 502–509 (1949)
Samuels C. L.: The infimum in the metric Mahler measure. Canad. Math. Bull., 54, 739–747 (2011)
Samuels C. L.: A collection of metric Mahler measures. J. Ramanujan Math. Soc., 25, 433–456 (2010)
Samuels C. L.: The parametrized family of metric Mahler measures. J. Number Theory, 131, 1070–1088 (2011)
Samuels C. L.: Metric heights on an Abelian group. Rocky Mountain J. Math., 44, 2075–2091 (2014)
Samuels C.L., Continued fraction expansions in connection with the metric Mahler measure, Monatsh. Math., 181 (2016), 907–935.
C. L. Samuels, Counting exceptional points for rational numbers associated to the Fibonacci sequence, Period. Math. Hungar., to appear.
A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith., 24 (1973), 385–399. Addendum, ibid., 26 (1975), 329–331.
Smyth C. J.: On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc., 3, 169–175 (1971)
Voutier P.: An effective lower bound for the height of algebraic numbers. Acta Arith., 74, 81–95 (1996)
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Samuels, C.L. Metric Mahler measures over number fields. Acta Math. Hungar. 154, 105–123 (2018). https://doi.org/10.1007/s10474-017-0770-y
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DOI: https://doi.org/10.1007/s10474-017-0770-y