sentences:  It should be remarked that Schmüdgen’s Positivstellensatz has a weaker assumption than Putinar’s Positivstellensatz, but the conclusion is also weaker.Suppose that formula_12.If the semialgebraic set formula_13 is compact, then each polynomial formula_14 that is strictly positive on formula_15 can be written as a polynomial in the defining functions of formula_16 with sums-of-squares coefficients, i.e.