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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

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