On the equivalence of algebraic approaches to the minimization of forms on the simplex

We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by Schmudgen-Putinar and by Polya about representations of positive polynomials in terms of sums of squares. We show that the two approaches yield, in fact, the same approximations. The same type of argument also permits to establish some representation results a la Polya for positive polynomials on semi-algebraic cones.