Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T21:42:46.085Z Has data issue: false hasContentIssue false
  • English
  • Français

An effective lower bound on the “Diophantine approximation” of algebraic functions by rational functions

Published online by Cambridge University Press:  26 February 2010

Charles F. Osgood
Charles F. Osgood
Affiliation:
Mathematics Research Center, Naval Research Laboratory, Washington, D.C., 20375.
Get access

Extract

The idea of obtaining bounds on the order of approximation of algebraic functions by rational functions apparently goes back to Maillet [8]. It is very easy, using an argument analogous to that used by Liouville for algebraic numbers, to see that if w(z) is any algebraic function of degree n ≥ 2 over Q[z] then for all pairs of polynomials r(z) and s(z) with s(z) ≢ 0 we have

Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 4 - 15
Copyright
Copyright © University College London 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Gill, B. P., “An Analogue for Algebraic Functions of the Thue-Siegel Theorem”, Annals of Mathematics, 31 (1930), 207–218.CrossRefGoogle Scholar
2. Hardy, G. H. and Ramanujan, S., “Asymptotic Formulae in Combinatory Analysis”, Proc. London Math. Soc., 17 (1918), 75–115.CrossRefGoogle Scholar
3. Hurwitz, A., “Sur le développement des fonctions satisfaisant à une équation différentielle algébrique”, Ann. de l ' École Norm. (3), 6, 327–332; Werke, Bd. 1, 295–298.CrossRefGoogle Scholar
4. Kolchin, E. R., “Rational Approximation to Solutions of Algebraic Differential Equations”, Proc. Amer. Math. Soc., 10 (1959), 238–244.CrossRefGoogle Scholar
5. Mahler, K., “On a Theorem of Liouville in Fields of Positive Characteristic”, Canadian Journal of Mathematics, 1 (1949), 397–400.CrossRefGoogle Scholar
6. Mahler, K., “Zur Approximation der Exponential funktion und des Logarithmus Teil I and Teil II”, Journal für die reine und angewandte Mathematik, 116, 118–150.Google Scholar
7. Mahler, K., “On formal power series as integrals of algebraic differential equations”, Rend. Ace. Naz. Lincei (8) 50, 36–49.Google Scholar
8. Maillet, E., Introduction a la théorie des nombres transcendants et des propriétes arithmétiques des fonctions (Paris, 1906).Google Scholar
9. Popken, J. (thesis), Über arithmetische Eigenschaften analytischer Funktionen Amsterdam, N.V. Noord-Hollandsche Uitgeversmaatschappij, 1935.Google Scholar
10. Uchiyama, S., “Rational approximations to algebraic functions”, J. Fac. Sci. Hokkaido Univ., Ser. 15, (1961), 173–192.Google Scholar