An effective version of Pólyaʹs theorem on positive definite forms Original Research Article

Given a real homogeneous polynomial F, strictly positive in the non-negative orthant, Pólyaʹs 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 Pólyaʹs 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. Pólyaʹs theorem has appeared before in constructive solutions of Hilbertʹs 17th problem for positive definite forms [4]. We also present a different procedure to do this kind of construction.