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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

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