An effective version of Pólya's theorem on positive definite forms

AbstractGiven a real homogeneous polynomial F, strictly positive in the non-negative orthant, Pólya's theorem says that for a sufficiently large exponent p the coefficients of F(x1,…,xn) · (x1 + … + xn)p are strictly positive. The smallest such p will be called the Pólya exponent of F. We present a new proof for Pólya's result, which allows us to obtain an explicit upper bound on the Pólya exponent when F has rational coefficients. An algorithm to obtain reasonably good bounds for specific instances is also derived.Pólya's theorem has appeared before in constructive solutions of Hilbert's 17th problem for positive definite forms [4]. We also present a different procedure to do this kind of construction