An algorithmic approach to Schmüdgen's Positivstellensatz

Schweighofer, Markus

An algorithmic approach to Schmüdgen's Positivstellensatz

Files

schmuedgen.pdf(995.54 KB)

Date

2002

Publication type

Journal article

Published in

Journal of Pure and Applied Algebra ; 166 (2002), 3. - pp. 307-319. - ISSN 0022-4049

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).

Subject (DDC)

520 Astronomy

Keywords

effective Positivstellensatz,strictly positive polynomials

Cite This

ISO 690

SCHWEIGHOFER, Markus, 2002. An algorithmic approach to Schmüdgen's Positivstellensatz. In: Journal of Pure and Applied Algebra. 166(3), pp. 307-319. ISSN 0022-4049. Available under: doi: 10.1016/S0022-4049(01)00041-X

BibTex

@article{Schweighofer2002algor-15653,
  year={2002},
  doi={10.1016/S0022-4049(01)00041-X},
  title={An algorithmic approach to Schmüdgen's Positivstellensatz},
  number={3},
  volume={166},
  issn={0022-4049},
  journal={Journal of Pure and Applied Algebra},
  pages={307--319},
  author={Schweighofer, Markus}
}

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