AN EFFECTIVE VERSION OF POLYAS THEOREM ON POSITIVE-DEFINITE FORMS

Given a real homogeneous polynomial F, strictly positive in the non-ne gative orthant, Polya's theorem says that for a sufficiently large exp onent p the coefficients of F(x(1),...,x(n)) (x(1)+...+x(n))(p) are st rictly positive. The smallest such p will be called the Polya exponent of F. We present a new proof for Polya's result, which allows us to o btain an explicit upper bound on the Polya exponent when F has rationa l coefficients. An algorithm to obtain reasonably good bounds for spec ific instances is also derived. Polya's theorem has appeared before in constructive solutions of Hilbert's 17th problem for positive definit e forms [4]. We also present a different procedure to do this kind of construction.