An algorithmic approach to Schmüdgen’s Positivstellensatz

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).