Inventiones mathematicae

, Volume 76, Issue 3, pp 365–384 | Cite as

A continuous, constructive solution to Hilbert's 17th problem

  • C. N. Delzell


Constructive Solution 17th Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Sem. Hamburg5, 100–15 (1927). (An English version of the main proof there is on p.289ff. of Jacobson's Lectures in Abstract Algebra3, 1964)Google Scholar
  2. Bochnak, J., Efroymson, G.: Real algebraic geometry and the 17th Hilbert problem. Math. Ann.251(3), 213–41 (1980)Google Scholar
  3. Bourbaki, N.: General Topology, Tome 1. Hermann, Addison-Wesley, Reading, Mass. 1966Google Scholar
  4. Brumfiel, G.: Partially Ordered Rings and Semi-Algebraic Geometry. Lecture Note Series of the London Math. Soc. Cambridge Univ. Press, Cambridge, 1979Google Scholar
  5. Choi, M.D., Lam, T.-Y.: Extremal positive semidefinite forms. Math. Ann.231, 1–18 (1977)Google Scholar
  6. Cohen, P.J.: Decision procedures for real andp-adic fields. Comm. in Pure & Applied Math.22, 131–51 (1969)Google Scholar
  7. Coste, M.F.: Recursive functions in topoi. Oberwolfach Tagungsberichte 1975Google Scholar
  8. Coste M., Coste-Roy, M.F.: Topologies for real algebraic geometry. Topos Theoretic Methods in Geometry. In: Kock, A. (ed.) Various Publications Series vol. 30. Mathematisk Institut, Aarhus Univ. 1979Google Scholar
  9. Daykin, D. E.: Thesis, Univ. of Reading, 1960 (unpublished); cited by Kreisel, A survey of proof theory. J. Symb. Logic33, 321–88 (1968)Google Scholar
  10. Delzell, C.N.: A constructive, continuous solution to Hilbert's 17th problem, and other results in semi-algebraic geometry, Ph.D. dissertation, Stanford Univ. 1980 (Univ. Microfilms International, Order No. 8024640). Cf. also Dissertation Abstracts International41, (no. 5) 1980, and AMS Abstracts2(1) (Jan. 1981), # 783-12-28Google Scholar
  11. Delzell, C.N.: Analytic version of Siegel's theorem on sums of squares, in preparation; preliminary abstract in AMS Abstracts3(2) (Jan. 1982a) #792-12-269Google Scholar
  12. Delzell, C.N.: A finiteness theorem for open semi-algebraic sets, with applications to Hilbert's 17th problem, Ordered Fields and Real Algebraic Geometry, Dubois, D.W., Recio, T. (eds.). Contemporary Math. Series, AMS, Providence, 1982b, pp. 79–97Google Scholar
  13. Delzell, C.N.: Case distinctions are necessary for representing polynomials as sums of squares. Proc. Herbrand Symp., Logic Coll. 1981, Stern, J. (ed.). North Holland, 1982c, pp. 87–103Google Scholar
  14. Delzell, C.N.: Continuous sums of squares of forms. L.E.J. Brouwer Centenary, Symp. Troelstra, A.S., van Dalen, D. (eds.). North Holland, 1982d, pp. 65–75Google Scholar
  15. Delzell, C.N.: Continuity., rationality, and minimality for sums of squares of linear forms, in preparation; preliminary abstract in AMS Abstracts4 (Jan. 1983) # 801-12-354Google Scholar
  16. Dries, L., van den: Some applications of a model-theoretic fact to (semi-)algebraic geometry. Indag. Math. in press (1984)Google Scholar
  17. Heilbronn, H.: On the representation of a rational as a sum of four, squares by means of regular functions. J. London Math. Soc.39, 72–6 (1964)Google Scholar
  18. Hilbert, D.: Über die Darstellung definiter Formen als Summe von Formenquadraten Math. Ann.32, 342–50 (1888); see also Ges. Abh. vol. 2, pp. 154–61 Berlin-Heidelberg-New York: Springer 1933Google Scholar
  19. Hilbert, D.: Grundlagen der Geometrie (Teubner, 1899); transl. by E.J. Townsend (Open Court Publishing Co. La Salle, IL, 1902); transl. by L. Unger from the tenth German edition (Open Court, 1971)Google Scholar
  20. Hilbert, D.: Mathematische Probleme, Göttinger Nachrichten (1900), pp. 253–97, and Archiv der Mathematik und Physik 3d ser.1, 44–53, 213–37 (1901). Transl. by M.W. Newson, Bull. Amer. Math. Soc.8, 437–79 (1902); reprinted in Mathematical Developments Arising from Hilbert Problems, Browder, F. (ed.), Proc. Symp. in Pure Math.28, Amer. Math. Soc., Providence, 1976, 1–34Google Scholar
  21. Hironaka, H.: Triangulations of semi-algebraic sets. Proc. Symp. in Pure Math.29, Amer. Math. Soc. Providence, 1975, pp. 165–85Google Scholar
  22. Hu, S.-T.: Theory of Retracts. Wayne State Univ. Press, Detroit, 1965Google Scholar
  23. Kreisel, G.: Mathematical significance of consistency proofs. J. Symb. Logic23, 155–82 (1958) (reviewed by A. Robinson, JSL31 128)Google Scholar
  24. Kreisel, G.: Review of Goodstein. Math. Reviews24A, #A1821, 336–7 (1962)Google Scholar
  25. Kreisel, G.: Review of Ershov. Zentralblatt374, 18–9, #02027 (1978)Google Scholar
  26. Kreisel, G., MacIntyre, A.: Constructive logic vs. algebraization L.E.J. Brouwer Cent. Symp. Troelstra, A.S., van Dalen, D. (eds.). North Holland, 1982, pp 217–60Google Scholar
  27. Lam, T.-Y.: The theory of ordered fields. Ring Theory and Algebra. III. McDonald, B. (ed.). New York: Marcel Bekker 1980Google Scholar
  28. McEnerney, J.: Trim stratification of semi-analytic sets. Manuscripta Math.25, (1) 17–46 (1978)Google Scholar
  29. Pfister, A.: Hilbert's 17th problem and related problems on definite forms, Mathematical Developments Arising from Hilbert Problems, Browder, F. (ed.) Proc. Symp. in Pure Math28, 483–489. Amer. Math. Soc. 1976Google Scholar
  30. Recio, T.: Actas de la IV Reunion de Matematicos de Expresion Latina. Mallorca, 1977Google Scholar
  31. Robinson, R.M.: Some definite polynomials which are not sums of squares of real polynomials. Notices Amer. Math. Soc.16, 554 (1969); Selected Questions in Algebra and Logic (Vol. dedicated to the memory of A.I. Mal'cev), Izdat. “Nauka” Sibirsk Otdel Novosibirsk 264-82 (1973); or Acad. Sci. USSR. (MR49 #2647)Google Scholar
  32. Stengle, G.: A Nullstellensatz and a Positivstellensatz for semi-algebraic geometry. Math. Ann.207, 87–97 (1974)Google Scholar
  33. Stengle, G.: Integral solution of Hilbert's 17th problem. Math. Ann.246, 33–39 (1979)Google Scholar
  34. Stout, L.N.: Topological properties of the real numbers object in a topos. Cahiers de Topologie et Géométrie Différentielle17(3), 295–376 (1976)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • C. N. Delzell
    • 1
  1. 1.Mathematics DepartmentLouisiana State UniversityBatou RougeUSA

Personalised recommendations