We consider the set of the power non-negative polynomials of severalvariables and its subset that consists of polynomials which can be representedas a sum of squares. It is shown in the classic work by D.Hilbert that it is aproper subset. Both sets are convex. In our paper we have made an attempt towork out a general approach to the investigation of the extremal elements ofthese convex sets. We also consider the class of non-negative rationalfunctions. The article is based on the following methods: 1.We investigatenon-negative trigonometrical polynomials and then with the help of the Calderontransformation we proceed to the power polynomials. 2.The way of constructingsupport hyperplanes to the convex sets is given in the paper.
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