Schwarz lemmas in several variables and arithmetical applications. (Lemmes de Schwarz en plusieurs variables et applications arithmétiques.) (French) Zbl 0452.10036

Moreau, Jean-Charles

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.]

Reviewer: T. A. Metzger (Pittsburgh)

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
30C80	Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

Schwarz lemmas; transcendental number theory; location of zeros; logarithms of algebraic numbers

Citations:

Zbl 0428.00008