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Mahler measures in a cubic field

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Abstract

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.

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References

  1. R. L. Adler, B. Marcus: Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. 20 (1979).

  2. D. W. Boyd: Inverse problems for Mahler’s measure. In: Diophantine Analysis. London Math. Soc. Lecture Notes Vol. 109 (J. Loxton and A. van der Poorten, eds.). Cambridge Univ. Press, Cambridge, 1986, pp. 147–158.

    Google Scholar 

  3. D. W. Boyd: Perron units which are not Mahler measures. Ergod. Th. and Dynam. Sys. 6 (1986), 485–488.

    MATH  MathSciNet  Google Scholar 

  4. D. W. Boyd: Reciprocal algebraic integers whose Mahler measures are non-reciprocal. Canad. Math. Bull. 30 (1987), 3–8.

    MATH  MathSciNet  Google Scholar 

  5. J. D. Dixon, A. Dubickas: The values of Mahler measures. Mathematika 51 (2004), 131–148.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Dubickas: Mahler measures close to an integer. Canad. Math. Bull. 45 (2002), 196–203.

    MATH  MathSciNet  Google Scholar 

  7. A. Dubickas: On numbers which are Mahler measures. Monatsh. Math. 141 (2004), 119–126.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Dubickas: Mahler measures generate the largest possible groups. Math. Res. Lett 11 (2004), 279–283.

    MATH  MathSciNet  Google Scholar 

  9. A.-H. Fan, J. Schmeling: ε-Pisot numbers in any real algebraic number field are relatively dense. J. Algebra 272 (2004), 470–475.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. H. Lehmer: Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 461–479.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Salem: Algebraic Numbers and Fourier Analysis. D. C. Heath, Boston, 1963.

    MATH  Google Scholar 

  12. A. Schinzel: Polynomials with Special Regard to Reducibility. Encyclopedia of Mathematics and its Applications Vol. 77. Cambridge University Press, Cambridge, 2000.

    MATH  Google Scholar 

  13. A. Schinzel: On values of the Mahler measure in a quadratic field (solution of a problem of Dixon and Dubickas). Acta Arith. 113 (2004), 401–408.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Waldschmidt: Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables. Springer-Verlag, Berlin-New York, 2000.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225, Vilnius, Lithuania

    Artūras Dubickas

  2. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania

    Artūras Dubickas

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  1. Artūras Dubickas
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Dubickas, A. Mahler measures in a cubic field. Czech Math J 56, 949–956 (2006). https://doi.org/10.1007/s10587-006-0069-6

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  • DOI: https://doi.org/10.1007/s10587-006-0069-6

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