Skip to main content
SpringerLink
Log in

A continuous and rational solution to Hilbert’s 17th problem and several cases of the Positivstellensatz

  • Conference paper
Computational Algebraic Geometry

Part of the book series: Progress in Mathematics ((PM,volume 109))

Abstract

From the Positivstellensatz we construct a continuous and rational solution for Hilbert’s 17th problem and for several cases of the Positivstellensatz. The solutions are obtained using an especially simple method.

Supported by NSF, the Louisiana Board of Regents Research and Development Program (Education Quality Support Fund), and the Alexander von Humboldt Foundation.

Partially supported by CICyT PB 89/0379/C02/01 and Esprit/Bra 6846 (Posso).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Artin E.: Uber die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Sem. Hamburg, 5 (1927), 100–115.

    Article  MathSciNet  Google Scholar 

  2. Bishop E., Bridges D.: Constructive Analysis. Springer-Verlag (1985).

    Google Scholar 

  3. Bochnak J., Coste M. and Roy M.-F.: Géométrie Algébrique Réelle. Ergebnisse vol. 12, Springer-Verlag (1987).

    Google Scholar 

  4. Bradley M., Prestel A.: Representation of a real polynomial f(X) as a sum of 2m-th powers of rational functions. Ordered Algebraic Structures, J. Martinez, ed., Kluwer (1989), 197–207.

    Google Scholar 

  5. Daykin D.: Hilbert’s 17 th problem. Ph.D. Thesis, unpublished (1961).

    Google Scholar 

  6. Delzell C.N.: A finiteness theorem for open semialgebraic sets, with applications to Hilbert’s 17th problem. Ordered Fields and Real Algebraic Geometry, Contemp. Math. 8, AMS, D. Dubois and T. Recio, (eds.), Providence 1982, 79–97.

    Google Scholar 

  7. Delzell C.N.: Case distinctions are necessary for representing polynomials as sums of squares. Proc. Herbrand Symp., Logic Coll. 1981, J. Stern (ed.), Amsterdam-Oxford-New York 1982, 87–103.

    Google Scholar 

  8. Delzell C.N.: A continuous, constructive solution to Hilbert’s 17th problem. Inventiones Mathematicae 76 (1984), 365–384.

    Article  MATH  MathSciNet  Google Scholar 

  9. Delzell C.N.: On the Pierce-Birkhoff conjecture over ordered fields. Rocky Mountain Journal of Mathematics 19(3) (Summer 1989), 651–668.

    Article  MATH  MathSciNet  Google Scholar 

  10. Delzell C.N.: A sup-inf-polynomially varying solution to Hilbert’s 17th problem. AMS Abstracts 10(3), (Issue 63, April 1989), 208–209, #849-14-160.

    MathSciNet  Google Scholar 

  11. Delzell C.N.: On analytically varying solutions to Hilbert’s 17th problem. Submitted to Proc. Special Year in Real Algebraic Geometry and Quadratic Forms at UC Berkeley, 1990–1991, (W. Jacob, T.-Y. Lam, R. Robson, eds.), Contemporary Mathematics.

    Google Scholar 

  12. Delzell C.N.: On analytically varying solutions to Hilbert’s 17th problem. AMS Abstracts 12(1), (Issue 73, January 1991), page 47, #863-14-743.

    Google Scholar 

  13. Delzell C.N.: Continuous, piecewise-polynomial functions which solve Hilbert’s 17th problem. Submitted for publication (1992).

    Google Scholar 

  14. Dubois D.W.: A nullstellensatz for ordered fields. Arkiv for Mat. 8 (1969), 111–114.

    Article  Google Scholar 

  15. Efroymson G.: Local reality on algebraic varieties. Journal of Algebra 29 (1974), 113–142.

    Article  MathSciNet  Google Scholar 

  16. González-Vega L., Lombardi H.: A Real Nullstellensatz and Positivstellensatz for the Semipolynomials over an Ordered Field. Submitted to the Journal of Pure and Applied Algebra (1992).

    Google Scholar 

  17. Hilbert D.: Mathematische Probleme. Göttinger Nachrichten (1900), 253–297, and Archiv der Math. u. Physik (3rd ser.) 1 (1901), 44–53, 213–237.

    Google Scholar 

  18. [Kre1]_Kreisel G.: Some uses of metamathematics. British J. Phil. Sci. 7(26) (August 1956), 161–173.

    Article  Google Scholar 

  19. [Kre2]_Kreisel G.: Sums of squares. Summaries of Talks Presented at the Summer Institute in Symbolic Logic in 1957 at Cornell Univ., Institute Defense Analyses, Princeton (1960), 313–320.

    Google Scholar 

  20. [Kre3]_Kreisel G.: Review of Goodstein, MR 24A, 336–337, #A1821 (1962).

    MathSciNet  Google Scholar 

  21. [Kre4]_Kreisel G.: Review of Ershov, Zbl. 374, 02027 (1978).

    Google Scholar 

  22. Krivine J.L.: Anneaux préordonnés. Journal d’Analyse Mathématique 12 (1964), 307–326.

    Article  MATH  MathSciNet  Google Scholar 

  23. [Lom1]_Lombardi H.: Effective real Nullstellensatz and variants. Effective Methods in Algebraic Geometry. Editors T. Mora and C. Traverso. Progress in Mathematics 94 (1991), 263–288, Birkhauser. Detailed French version in Théorème effectif des zéros réel et variantes (avec une majoration explicite des degrés), Memoire d’habilitation (1990).

    Google Scholar 

  24. [Lom2]_Lombardi H.: Une étude historique sur les problèmes d’effectivité en algèbre réelle. Memoire d’habilitation (1990).

    Google Scholar 

  25. [Lom3]_Lombardi H.: Une borne sur les degrés pour le théorème des zéros réel effectif. Proceedings of the conference on Real Algebraic Geometry held in La Turballe (1991). To appear as Lecture Notes in Mathematics.

    Google Scholar 

  26. Lombardi H., Roy M.-F.: Théorie constructive élémentaire des corps ordonnés. Publications Mathematiques de Besançon, Théorie de Nombres, 1990–1991. English version in: Constructive elementary theory of ordered fields, Effective Methods in Algebraic Geometry. Editors T. Mora and C. Traverso. Progress in Mathematics 94 (1991), 249–262, Birkhauser.

    Google Scholar 

  27. Mines R., Richman F., Ruitenburg W.: A Course in Constructive Algebra. Universitext, Springer-Verlag (1988).

    Google Scholar 

  28. [Pre1]_Prestel A.: Lectures on Formally Real Fields. IMPA Lecture Notes 22, Rio de Janeiro 1975; reprinted in Lecture Notes in Mathematics 1093 (1983), Springer-Verlag.

    Google Scholar 

  29. [Pre2]_Prestel A.: Continuous representations of real polynomials as sums of 2 m -th powers. Math. Forschungsinst. Oberwolfach Tagungsbericht (Reelle algebraische Geometrie, June 10–16, 1990) 25 (1990), 18.

    Google Scholar 

  30. Risler J.-J.: Une caractérisation des idéaux des variétés algébriques réelles. C.R.A.S. Paris, Série A, 271 (1970), 1171–1173.

    Google Scholar 

  31. [Rob1]_Robinson A.: On ordered fields and definite forms. Math. Ann. 130 (1955), 257–271.

    Article  MATH  Google Scholar 

  32. [Rob2]_Robinson A.: Further remarks on ordered fields and definite forms. Math. Ann. 130 (1956), 405–409.

    Article  MATH  MathSciNet  Google Scholar 

  33. Schwartz N.: Piecewise-polynomial functions. Submitted.

    Google Scholar 

  34. [Sco1]_Scowcroft P.: A transfer theorem in constructive real algebra. Ann. Pure and Appl. Logic 40(1), 29–87 (1988).

    Article  MathSciNet  Google Scholar 

  35. [Sco2]_Scowcroft P.: Some continuous Positivstellensatze. Journal of Algebra 124 (1989), 521–532.

    Article  MATH  MathSciNet  Google Scholar 

  36. Solernó P.: Effective Lojasiewicz inequalities in semialgebraic geometry. Applicable Algebra in Engineering, Communication and Computing 2(1) (1991), 1–14.

    Article  MATH  Google Scholar 

  37. Stengle G.: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87–97.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Charles N. Delzell

  2. Dept. Matemáticas, Universidad de Cantabria, Santander, 39071, Spain

    Laureano González-Vega

  3. Lab. de Mathématiques, Université de Franche-Comté, Besançon, 25030, France

    Henri Lombardi

Authors
  1. Charles N. Delzell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laureano González-Vega
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Henri Lombardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. U.F.R. Faculté des Sciences Mathématiques, Université de Nice-Sophia Antipolis, 06108, Nice Cedex 2, France

    Frédéric Eyssette  & André Galligo  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Birkhäuser Boston

About this paper

Cite this paper

Delzell, C.N., González-Vega, L., Lombardi, H. (1993). A continuous and rational solution to Hilbert’s 17th problem and several cases of the Positivstellensatz. In: Eyssette, F., Galligo, A. (eds) Computational Algebraic Geometry. Progress in Mathematics, vol 109. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2752-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2752-6_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7652-4

  • Online ISBN: 978-1-4612-2752-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation