Use of complex conjugation in the transcendence of values of the usual exponential function. (Utilisation de la conjugaison complexe dans l’étude de la transcendance de valeurs de la fonctions exponentielle usuelle.) (French) Zbl 1071.11046

Diaz, Guy

J. Théor. Nombres Bordx. 16, No. 3, 535-553 (2004).

The starting point of the author is the following conjecture of his own: if \(u\) is a nonzero complex number, the modulus \(| u| \) of which is algebraic, then \(e^{u}\) is transcendental. This generalizes the Hermite Lindemann Theorem where the assumption is that \(u\) itself is algebraic. In connection with this conjecture, the author shows how complex conjugation can be used in transcendence theory. One of the earliest occurrence of such considerations can be traced back to K. Ramachandra’s papers [”Contributions to the theory of transcendental numbers. I, II.” Acta Arith. 14, 65–72, 73–88 (1968; Zbl 0176.33101)]. As an illustration the author considers the following statement: if \(\alpha\) is an algebraic number not in \(\mathbb R\) nor in \(i\mathbb R\), then for all \(v\in\mathbb R\) the number \(e^{v\alpha}\) is transcendental. He first notices that this statement is a consequence of Gel’fond Schneider Theorem on the transcendence of \(a^{b}\), hence that it can be proved (using Schneider’s method) without using differential equations. Next he gives a full proof where no derivative is involved and no analytic tool like Schwarz Lemma is used. He concludes by considering further consequences of Schanuel’s Conjecture.

Reviewer: Michel Waldschmidt (Paris)

MSC:

11J81	Transcendence (general theory)
11J91	Transcendence theory of other special functions

Keywords:

transcendence; Hermite; Lindemann; Gel’fond; Schneider; exponential function; Schanuel’s conjecture; Schwarz Lemma; complex conjugation

Citations:

Zbl 0176.33101

References:

[1]	G. Diaz, La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire. J. Théorie des Nombres de Bordeaux 9 (1997), 229-245. · Zbl 0887.11030
[2]	A.0. Gel’Fond, Ju.V. Linnik, Méthodes élémentaires dans la théorie analytique des Nombres. Monographies internationales de Mathématiques modernes vol. 6, Gauthier-Villars, Paris, 1965. · Zbl 0125.29604
[3]	M. Laurent, Sur quelques résultats récents de transcendance. Astérisque n° 198-200 (1991), 209-230. · Zbl 0762.11027
[4]	D. Roy, An arithmetic criterion for the values of the exponential function. Acta Arith. XCVII. 2 (2001), 183-194. · Zbl 0981.11025
[5]	M. Waldschmidt, Une méthode élémentaire dans la théorie des nombres transcendants I, II. Séminaire DPP, n° G1 et G5 (1972/73). · Zbl 0325.10020
[6]	M. Waldschmidt, Nombres transcendants. Lecture Notes in Mathematics vol. 402, Springer-Verlag, Berlin-New York, 1974. · Zbl 0302.10030
[7]	M. Waldschmidt, Diophantine approximation on linear algebraic groups. Grundlerhen der mathematischen Wissenschaften, vol. 326, Springer-Verlag, Berlin-Heidelbeg-New York, 2000. · Zbl 0944.11024