J Austral Math Soc Ser A 54 pp111--127, 1993.

Geometric and Arithmetic Postulation of the Exponential Function

J. Pila

Abstract

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 hand.

1991 AMS Subject Classification: 11J81

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Authors

J. Pila

Columbia University, New York, NY 10027, USA.