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

• Yann Bugeaud
Conference paper
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

11J04

## References

1. 1.
Baker, R.C.: On approximation with algebraic numbers of bounded degree. Mathematika 23, 18–31 (1976)
2. 2.
Bugeaud, Y.: Mahler’s classification of numbers compared with Koksma’s. Acta Arith. 110, 89–105 (2003)
3. 3.
Bugeaud, Y.: Mahler’s classification of numbers compared with Koksma’s, III. Publ. Math. (Debrecen) 65, 305–316(2004)
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)
6. 6.
Evertse, J.-H.: Distances between the conjugates of an algebraic number. Publ. Math. (Debrecen) 65, 323–340 (2004)
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)
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)
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)
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)