An algorithmic approach to SchmüdgenʹsPositivstellensatz

We present a new proof of SchmüdgenʹsPositivstellensatz concerning the representation of polynomials that are strictly positive on a compact basic closed semialgebraic subset Sof . 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 (minus the origin).