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An algorithmic approach to Schmüdgen's Positivstellensatz

✍ Scribed by Markus Schweighofer

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
117 KB
Volume
166
Category
Article
ISSN
0022-4049

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✦ Synopsis

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

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