Entire functions with values in a number field. (Fonctions entières à valeurs dans un corps de nombres.) (English. French summary) Zbl 1219.11045

Ably, Mohammed

Bull. Soc. Math. Fr. 139, No. 2, 243-270 (2011).

Let \(\Gamma\) be an additive subgroup of maximal rank in a number field \(k\). The author proves that any entire function on \(\Gamma\) with integer values in a finite extension of \(k\) which has both sufficiently slow analytical and arithmetical growth is a polynomial. This result extends a well-known theorem of Pólya. Moreover, it is proved that this result is optimal up to a constant. The proof uses a transcendental method. The construction of suitable interpolation polynomials plays an important role.

Reviewer: Maurice Mignotte (Strasbourg)

MSC:

11C08	Polynomials in number theory
11H06	Lattices and convex bodies (number-theoretic aspects)
30D15	Special classes of entire functions of one complex variable and growth estimates

Keywords:

number field; entire function; polynomial; lattice; interpolation