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Polya’s theorem for non-entire functions. (English) Zbl 0553.32002
Let $${\mathbb{Z}}$$ be the integers and $${\mathbb{N}}$$ the positive integers. A famous theorem of Polya states that if f(z) is an entire function for $$z\in {\mathbb{C}}$$ and $$| f(z)| \leq C\cdot 2^{| z|},$$ $$f({\mathbb{N}})\subset {\mathbb{Z}},$$ then f(z) is necessarily a polynomial with rational coefficients. The author generalizes this theorem to functions of several complex variables defined and of exponential type in a product of half spaces in the following way: Suppose that f(z) is holomorphic in $$D=\{z\in {\mathbb{C}}^ n:Re z_ i>0,\quad i=1,...,n\}$$ and that a(z) is a convex function real homogeneous of order 1 such that for every $$\epsilon >0$$, there exists a constant $$C_{\epsilon}>0$$ for which $$| f(z)| \leq C_{\epsilon}\exp a(z)$$ for $$Re z_ i\geq \epsilon$$, $$i=1,...,n$$. Let $$L=\{\xi \in {\mathbb{C}}^ n:Re<\xi,z>\leq a(z)\quad for\quad all\quad z\in D\},$$ and suppose that $$| \exp \xi_ i-1| <1,\quad i=1,...,n$$ for all $$\xi\in L$$. Then if $$f({\mathbb{N}}^ n)\subset {\mathbb{Z}},$$ f(z) is a polynomial with rational coefficients.
Reviewer: L.Gruman

##### MSC:
 32A10 Holomorphic functions of several complex variables 30C10 Polynomials and rational functions of one complex variable 32A30 Other generalizations of function theory of one complex variable