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Constructive proofs of some positivstellensätze for compact semialgebraic subsets of $\mathbb{R}^d$

In a broad sense, positivstellensätze are results about representations of polynomials which are strictly positive on a given set. We give constructive and, to a large extent, elementary proofs of some known positivstellensätze for compact semialgebraic subsets of $\mathbb{R}^d$. The presented proofs extend and simplify arguments of Berr, Wörmann (2001) and Schweighofer (2002, 2005).