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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
##### Keywords:
number field; entire function; polynomial; lattice; interpolation
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