Lithuanian Mathematical Journal

, Volume 59, Issue 1, pp 48–53 | Cite as

Mahler measure of a difference of two conjugates*

  • Artūras DubickasEmail author


Suppose that an algebraic number β of degree d = n(n − 1) over ℚ is expressible by the difference of two conjugate algebraic integers α1α2 of degree n, namely, β = α1− α2. We prove that then there exists a constant c > 1, which depends on \( \overline{\mid \alpha \mid } \) = max1≤inαi∣ only, such that M(β)1/d > c.


Mahler’s measure difference of two conjugates Galois group Bogomolov property 





  1. 1.
    F. Amoroso, Mahler measure on Galois extensions, Int. J. Number Theory, 14(6):1605–1617, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. Amoroso and R.A. Dvornicich, A lower bound for the height in Abelian extensions, J. Number Theory, 80(2):260–272, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    F. Amoroso and D. Masser, Lower bounds for the heights in Galois extensions, Bull. Lond. Math. Soc., 48(6):1008–1012, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F. Amoroso, I. Pritsker, C. Smyth, and J. Vaaler, Appendix to report on BIRS workshop 15w5054 on the geometry, algebra and analysis of algebraic numbers: Problems proposed by participants, available from:
  5. 5.
    G. Baron, M. Drmota, and M. Skałba, Polynomial relations between polynomial roots, J. Algebra, 177(3):827–846, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Math. Monogr, Vol. 4, Cambridge Univ. Press, Cambridge, 2006.Google Scholar
  7. 7.
    A. Dubickas, On the average difference between two conjugates of an algebraic number, Lith. Math. J., 35(4):328–332, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Dubickas, On numbers which are differences of two conjugates of an algebraic integer, Bull. Aust. Math. Soc., 65(3):439–447, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Dubickas, Additive Hilbert’s theorem 90 in the ring of algebraic integers, Indag. Math., New Ser., 17(1):31–36, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. Dubickas and C.J. Smyth, Variations on the theme of Hilbert’s theorem 90, Glasg. Math. J., 44(3):435–441, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Höhn and N.P. Skoruppa, Un résultat de Schinzel, J. Théorie Nombres Bordeaux, 5(1):185, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Langevin, Calculs explicites de constantes de Lehmer, in Groupe de Travail en Théorie Analytique et Élémentaire des Nombres, 1986–1987, Publ. Math. Orsay, Vol. 88-01, Orsay, 1988, pp. 52–68.Google Scholar
  13. 13.
    D.H. Lehmer, Factorization of certain cyclotomic functions, Ann. Math. (2), 34(3):461–479, 1933.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Mignotte, Sur un théorème de M. Langevin, Acta Arith., 54(1):81–86, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith., 24:385–399, 1973. Addendum, Acta Arith., 26:329–331, 1974–1975.Google Scholar
  16. 16.
    C. Smyth, The Mahler measure of algebraic numbers: A survey, in Number Theory and Polynomials, Lond. Math. Soc. Lect. Note, Ser., Vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 322–349.Google Scholar
  17. 17.
    T. Zaïmi, On the integer form of the Additive Hilbert’s Theorem 90, Linear Algebra Appl., 390:175–181, 2004.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVilnius UniversityVilniusLithuania

Personalised recommendations