Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T17:36:24.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric and arithmetic postulation of the exponential function

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

J. Pila
J. Pila
Affiliation:
Columbia UniversityNew York NY 10027, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper presents new proofs of some classical transcendence theorems. We use real variable methods, and hence obtain only the real variable versions of the theorems we consider: the Hermite-Lindemann theorem, the Gelfond-Schneider theorem, and the Six Exponentials theorem. We do not appeal to the Siegel lemma to build auxiliary functions. Instead, the proof employs certain natural determinants formed by evaluating n functions at n points (alternants), and two mean value theorems for alternants. The first, due to Pólya, gives sufficient conditions for an alternant to be non-vanishing. The second, due to H. A. Schwarz, provides an upper bound.

MSC classification

Secondary: 11J81: Transcendence (general theory)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 111 - 127
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bombieri, E. and Pila, J., ‘The number of integral points on arcs and ovals’, Duke Math. J. 59 (1989), 337–357.CrossRefGoogle Scholar
[2]Lang, S., Introduction to transcendental numbers (Addison-Wesley, Palo Alto, 1966).Google Scholar
[3] Laurent, M., ‘Sur quelques resultats recents de transcendence’, in: Journeés arithmétiques de Luminy/89, Astérisque, to appear.Google Scholar
[4]Laurent, M., letter dated 12 June 1991.Google Scholar
[5]Pila, J., ‘Geometric postulation of a smooth function and the number of rational points’, Duke Math. J. 63 (1991), 449–463.CrossRefGoogle Scholar
[6]Pólya, G., ‘On the mean value theorem corresponding to a given linear homogeneous differential equation’, Trans. Amer. Math. Soc. 24 (1922), 312–324;Google Scholar
also Collected Papers, vol. 3 (M.I.T. Press, Cambridge, 1984) pp. 96–108.Google Scholar
[7]Pólya, G., and Szegő, G., Problems and Theorems in Analysis (Springer-Berlin, 1976).CrossRefGoogle Scholar
[8]Posse, C., ‘Sur le terme complémentaire de la formule de M. Tchebychef donnant l'expression approchée d'une intégrale définie par d'autres prises entre les mêmes limites’, Bull. Sci. Math. (2) 7 (1883), 214–224.Google Scholar
[9]Ramachandra, K., ‘Contributions to the theory of transcendental numbers’, Acta Arith. 14 (1968), 65–68.CrossRefGoogle Scholar
[10]Schendel, L., ‘Mathematische Miscellen. III. Das alternierende Exponential-differenzproduct’, Zeitschrift fur Math. u. Phys. 38 (1891), 84–87.Google Scholar
[11]Schwarz, H. A., ‘Verallgemeinerung eines analytischen Fundamentalsatzes’, Annali di Mat. (2) 10 (1880), 129–136.CrossRefGoogle Scholar
Also Gesammelte Mathematische Abhandlungen Bd. 2 (Julius Springer, Berlin 1890), pp. 296–302.Google Scholar
[12]Stieltjes, T. J., ‘Sur une généralisation de la formule des accroissements finis’, Bull. Soc. Math. de France 16 (1887), 100–113.Google Scholar
[13]Swinnerton-Dyer, H. P. F., ‘The number of lattice points on a convex curve’, J. Number Theory 6 (1974), 128–135.CrossRefGoogle Scholar
[14]Waldschmidt, M., ‘On the transcendence methods of Gelfond and Schneider in several variables’, in: New Advances in Transcendence Theory (ed. Baker, ) (C. U. P. Cambridge, 1988).Google Scholar
[15]Waldschmidt, M., ‘Transcendence des valeurs de la fonction exponentielle (course notes, Univ. ParisVI, 1990).Google Scholar
[16]Waldschmidt, M., ‘Transcendence problems in several variables, plenary address, third conference the Canadian number theory association, Queen's University, 1991.Google Scholar
[17]Weihrauch, K., ‘Ueber eine algebraische Determinante mit eigenthümlichem Bildungsgesetz der Elemente’, Zeitschrift fur Math. u. Phys. 36(1889), 34–40.Google Scholar