An algorithmic approach to Schmüdgen's Positivstellensatz

Abstract

We present a new proof of Schmüdgen's Positivstellensatz concerning the representation of polynomials f∈ R[X 1,…,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 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 R d (minus the origin).