An effective version of Pólya's theorem on positive definite forms
Let \(S= \{x\in\mathbb R^n\mid x_1\geq 0, \ldots, x_n\geq 0,\, \sum x_i>0\}\). A classical result by Pólya asserts that any homogeneous polynomial \(F\in\mathbb R [X_1, \ldots, X_n]\) with \(F(x)>0\) for \(x\in S\) can be represented as a quotient \({G \over (X_1+ \cdots + X_n)^p}\) where all coefficients of \(G\in\mathbb R [X_1, \ldots, X_n]\) are positive. The least exponent \(p\) having this property is called the Pólya exponent. The authors prove a sharpened version of Pólya's theorem which supplies an upper bound for the Pólya exponent. If the coefficients of \(F\) are all rational then the upper bound involves the number of variables, the degree of \(F\) and the size of the coefficients of \(F\). The result is applied to find a representation of a positive definite polynomial \(F\in \mathbb R [X_1, \ldots, X_n]\) as a quotient of two sums of squares \(G,H\in \mathbb R [X_1, \ldots, X_n]\) (Hilbert's 17th problem).
Pólya's theorem on positive definite forms
scientific article; zbMATH DE number 897489