Transcendence Proofs in One Variable
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The present chapter is an introduction to the method which will be developed in this book. However, we consider here only functions of a single variable. Our aim is to prove the theorems of Hermite-Lindemann and Gel’ fond-Schneider by means of the alternants or interpolation determinants of M. Laurent [Lau 1989]. The real case of these two theorems (§§ 2.3 and 2.4) is easier, thanks to an estimate, due to G. Pólya (Lemma 2.2), for the number of real zeroes of real exponential polynomials. For the complex (i.e. general) case (§§ 2.5 and 2.6), another type of zero estimate, due to Y. V. Nesterenko, will be used. In the first section we explain the method, and in the second one we introduce a few auxiliary lemmas It should be pointed out that the proof of our transcendence criterion (Lemma 2.1, which rests on Liouville’s inequality) will be given only in the next chapter.
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