An algorithmic approach to Schmüdgen's Positivstellensatz

Abstract

We present a new proof of Schmüdgen's Positivstellensatz concerning the representation of polynomials ƒ ∈ 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 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 ƒ 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).