An extension of Pólya’s theorem on power series with integer coefficients
Raphael M. Robinson · Transactions of the American Mathematical Society · 1968
The theorem which we shall extend is the following.Pólya's Theorem.Let E be a bounded closed set which is symmetric to the real axis.Suppose that the complement of E is a region G. Suppose that f(z) is regular in G and near oo has the expansionwith integer coefficients.If the transfinite diameter of E is less than 1, thenf(z) must be rational.This was proved first by Pólya [4] for the case in which G is simply connected.A general proof was given later by Pólya [5, p. 703A more precise conclusion is that/(z) can be expressed in the form P(z)/Q(z), where P(z) and Q(z) are relatively prime polynomials with integer coefficients, and Q(z) has its leading coefficient equal to 1. Thus the poles off(z) can occur only at sets of conjugate algebraic integers lying in F. By a theorem of Fekete [2], there are only a finite number of such algebraic integers when the transfinite diameter of F is less than 1.Pólya's theorem is an extension of a theorem of Carlson [1], according to which a function/(z) of the same form which is regular for \z\ > 1 must either be rational or have the unit circle as a natural boundary.(This was stated originally for power series regular inside the unit circle.)In [4], Pólya also proved the converse for the case in which G is simply connected.That is, if the transfinite diameter of E is at least 1, then there exist nonrational functions f(z) whose expansions at oo have integer coefficients; indeed, there are a nondenumerable infinity of such functions.I have not seen a proof of the converse in general.One will be given in this paper as part of our discussion of the extended Pólya theorem.Fekete [2] proved that the transfinite diameter of E is less than 1 if and only if there exists a polynomial h(z) = zr + Ar_xzr-1+---+A0 such that \h(z)\ < 1 on F. We may take the coefficients to be real.Indeed, using a