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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
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Abstract

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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