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Mahler’s Classification of Numbers Compared with Koksma’s, II

  • Yann Bugeaud
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

Mahler [8], in 1932, and Koksma [7], in 1939, introduced two related measures of the degree of approximation of a complex transcendental number ξ by algebraic numbers. Following Mahler [8], for any integer n ≥ 1, we denote by wn(ξ) the supremum of the exponents w for which
$$ 0 < \left| {P\left( \xi \right)} \right| < H\left( P \right)^{ - \omega } $$
has infinitely many solutions in integer polynomials P(X) of degree at most n. Here, H(P) stands for the naïve height of the polynomial P(X), that is, the maximum of the absolute values of its coefficients. Following Koksma [7], for any integer n ≥ 1, we denote by w n ξ the supremum of the exponents w for which
$$ 0 < \left| {\xi - \alpha } \right| < H\left( \alpha \right)^{ - \omega - 1} $$
has infinitely many solutions in complex algebraic numbers α of degree at most n. Here, H(α) stands for the naïve height of α, that is, the naïve height of its minimal defining polynomial over Z. Clearly, the functions w1 and w 1 * coincide.

Keywords

Classification of complex numbers roots of polynomial 

2000 Mathematics subject classification

11J04 

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References

  1. 1.
    Baker, R.C.: On approximation with algebraic numbers of bounded degree. Mathematika 23, 18–31 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bugeaud, Y.: Mahler’s classification of numbers compared with Koksma’s. Acta Arith. 110, 89–105 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bugeaud, Y.: Mahler’s classification of numbers compared with Koksma’s, III. Publ. Math. (Debrecen) 65, 305–316(2004)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bugeaud, Y: Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics, vol. 160. Cambridge University Press, Cambridge (2004)Google Scholar
  5. 5.
    Bugeaud, Y., Mignotte, M.: On the distance between roots of an integer polynomial. Proc. Edinb. Math. Soc. 47, 553–556 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Evertse, J.-H.: Distances between the conjugates of an algebraic number. Publ. Math. (Debrecen) 65, 323–340 (2004)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Koksma, J.F.: Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen. Monatsh. Math. Phys. 48, 176–189 (1939)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mahler, K.: Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II. J. Reine Angew. Math. 166, 118–150(1932)Google Scholar
  9. 9.
    Roy, D.: Approximation to real numbers by cubic algebraic numbers, II. Ann. Math. 158, 1081–1087 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Schmidt, W.M.: T-numbers do exist. In: Symposia Mathematica su Teoria dei Numeri, Istituto Nazionale di Alta Mathematica, Rome 1968. Symp. Math., 4, pp. 3–26. Academic Press, London (1970)Google Scholar
  11. 11.
    Schmidt, W.M.: Mahler’s T-numbers. In: 1969 Number Theory Institute. Proc. Symp. Pure Math., vol. 20, pp. 275–286. American Mathematical Society, Providence (1971)Google Scholar
  12. 12.
    Schönhage, A.: Polynomial root separation examples. J. Symb. Comput. 41, 1080–1090 (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Sprindžuk, V.G.: Mahler’s Problem in Metric Number Theory. American Mathematical Society, Providence (1969)Google Scholar
  14. 14.
    Wirsing, E.: Approximation mit algebraischen Zahlen beschränkten Grades. J. Reine Angew. Math. 206, 67–77(1961)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Yann Bugeaud
    • 1
  1. 1.U. F. R. de mathématiquesUniversité Louis PasteurStrasbourgFrance

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