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 P?3lya, gives sufficient conditions for an alternant to be non-vanishing. The second, due to H. A. Schwarz, provides an upper bound.
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机译：本文提出了一些经典超越定理的新证明。我们使用实变量方法，因此仅获得我们考虑的定理的实变量版本：Hermite-Lindemann定理，Gelfond-Schneider定理和六指数定理。我们不呼吁Siegel引理建立辅助功能。取而代之的是，证明使用某些自然行列式，这些自然行列式是通过在n个点评估n个函数（交替）而形成的，并为交替子提供了两个平均值定理。由于P 3lya，第一个给出了足够的条件以使交流电不消失。第二个，由于H. A. Schwarz，提供了一个上限。
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