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

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
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