An algorithmic approach to Schmüdgen's Positivstellensatz

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Abstract

We present a new proof of Schmüdgen's Positivstellensatz concerning the representation of polynomials f∈R[X1,…,Xd] that are strictly positive on a compact basic closed semialgebraic subset S of Rd. Like the two other existing proofs due to Schmüdgen and Wörmann, our proof also applies the classical Positivstellensatz to non-constructively produce an algebraic evidence for the compactness of S. But in sharp contrast to Schmüdgen and Wörmann we explicitly construct the desired representation of f from this evidence. Thereby we make essential use of a theorem of Pólya concerning the representation of homogeneous polynomials that are strictly positive on an orthant of Rd (minus the origin).

MSC

12Y05
13P99
14P10
14Q20

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