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Schwarz lemmas in several variables and arithmetical applications. (Lemmes de Schwarz en plusieurs variables et applications arithmétiques.) (French) Zbl 0452.10036
Sémin. P. Lelong - H. Skoda, Analyse, Années 1978/79, Lect. Notes Math. 822, 174-190 (1980).
The author proves some “Schwarz type lemmas” and applies them to transcendental number theory. In particular, a problem of D. Masser on the location of the zeros of special polynomials is proved. Too many results are stated and proved to enumerate in this short review but to give one a flavor of the applications we shall state one such result (Theorem 3.2 of the paper):
Let \(m\) be a non-negative integer and suppose \(\beta_0\) is an algebraic number such that \(\beta_0\) and \(m\) are not both zero. Let \(\beta_1,\ldots,\beta_n\) be algebraic real numbers such that \(1, \beta_1,\ldots,\beta_n\) are \(\mathbb Q\) linearly independent. Let \(l_1,\ldots, l_n\) be non-zero logarithms of algebraic numbers then \(\exp(\beta_0+\beta_1l_1+\ldots+\beta_nl_n)\) is transcendental. -
This theorem implies a result of A. Baker in the real case and generalizes a result due to M. Waldschmidt in the case \(\beta_0 = 0\) and \(m\ne 0\).
[For the entire collection see Zbl. 428.00008.]

MSC:
11J91 Transcendence theory of other special functions
11J86 Linear forms in logarithms; Baker’s method
32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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