A \(q\)-analogue of the Fukazawa-Gel’fond-Gramain theorem. (Un \(q\)-analogue du théorème de Fukazawa-Gel’fond-Gramain.) (French. English summary) Zbl 1294.30056

Summary: Let \(q\in \mathbb Z\) such that \(|q|\geqslant 2\). In this note, we show that if \(f\) is an entire function such that \(f(q^{ n }+iq^{ m })\in \mathbb Z[i]\) for \(n,m\in \mathbb N\), and if \(f\) is of sufficiently slow growth, then \(f\) is a polynomial.


30D15 Special classes of entire functions of one complex variable and growth estimates
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[1] Bundschuh (P.).— Arithmetische Eigenschaften ganzer Funktionen mehrerer Variablen. J für die reine Angew Math, 313, p. 116-132 (1980). · Zbl 0411.10009
[2] Bundschuh (P.).— A theorem of Gel’fond via Schneider’s method. In : New trends in probability and statistics, vol II (Palanga 1991) (ed by F.Schweiger and E. Manstavicius), 9-15. VSP, Utrecht (1992). · Zbl 0774.11033
[3] Fukazawa (S.).— Uber ganzwertige ganze Funktionen. Tôhoku Math J, 27, p. 41-52 (1926). · JFM 52.0318.01
[4] Gel’fond (A.).— Sur les propriétés arithmétiques des fonctions entières. Tôhoku Math J, 30, p. 280-285 (1929). · JFM 55.0116.01
[5] Gel’fond (A.).— Sur les fonctions entières qui prennent des valeurs entières dans les points \(β ^n\). Mat Sb, 40, p. 42-47 (1933). · Zbl 0007.12102
[6] Gramain (F.).— Sur le théorème de Fukasawa-Gel’fond. Invent. Math, 63, no. 3, p. 495-506 (1981). · Zbl 0461.10028
[7] Les nombres transcendants. Mém. Soc. Math. France (N.S.), No. 13 (1984).
[8] Polya (G.).— Uber ganzwertige ganze Funktionen. Rend Circ Mat Palermo, 40, p. 1-16 (1915). · JFM 45.0655.02
[9] Ramis (J-P.).— About the growth of entire functions solutions of linear algebraic \(q\)-difference equations. Ann. Fac. Sci. Toulouse Math, 6, no. 1, p. 53-94 (1992). · Zbl 0796.39005
[10] Waldschmidt (M.).— Polya’s theorem by Schneider’s method. Acta Math. Acad. Sci. Hungar. 31, no. 1-2, p. 21-25 (1978). · Zbl 0381.10029
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