Mathematische Zeitschrift

, Volume 220, Issue 1, pp 75–97 | Cite as

Uniform denominators in Hilbert's seventeenth problem

  • Bruce Reznick


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B1] E. Becker,Valuations and real places in the theory of formally real fields, Géométrie algébrique réelle et formes quadratiques, Proceedings, Rennes 1981 (J.-L.Colliot Thélène, M. Coste, L. Mahé et M.-F. Roy, eds.), Lecture Notes in Math. No. 959, Springer-Verlag, Berlin, Heidelberg, New York 1982, pp. 1–40Google Scholar
  2. [B2] E. Becker,The real holomorphy ring and sums of 2n-th powers, Géométrie algébrique réelle et formes quadratiques, Proceedings, Rennes 1981 (J.-L.Colliot Thélène, M. Coste, L. Mahé et M.-F. Roy, eds.), Lecture Notes in Math. No. 959, Springer-Verlag, Berlin, Heidelberg, New York 1982, pp. 139–181Google Scholar
  3. [B3] E. Becker and V. Powers,Sums of powers in rings and the real holomorphy ring, preprintGoogle Scholar
  4. [C1] M. D. Choi, T. Y. Lam and B. Reznick,Even symmetric sextics, Math. Z.195 (1987), 559–580MATHCrossRefMathSciNetGoogle Scholar
  5. [C2] W. K. Clifford,On syzygetic relations among the powers of linear quantics originally in Proc. Lond. Math. Soc., vol. 3, 1869, Paper XIV in Mathematical papers, Chelsea, New York, 1968Google Scholar
  6. [D1] P. J. Davis and P. Rabinowitz,Methods of numerical integration, 2nd ed., Academic Press, Orlando, 1984MATHGoogle Scholar
  7. [D2] J. A. de Loera and F. Santos,An effective version of Pólya's thoerem on positive definite forms, preprintGoogle Scholar
  8. [D3] C. N. Delzell,A constructive, continuous solution to Hilbert's 17th problem, and other results in semi-algebraic geometry, Ph.D. thesis, Stanford University, 1980Google Scholar
  9. [D4] C. N. Delzell,Case distinctions are necessary for representing polynomials as sums of squares, Proceedings of the Herbrand Symposium Logic Colloquium '81 (J. Stern, ed.), North-Holland, 1982, pp. 87–103Google Scholar
  10. [D5] C. N. Delzell,Continuous, piecewise-polynomial functions which solve Hilbert's 17th problem, J. Reine Angew. Math.440 (1993), 157–173MATHMathSciNetGoogle Scholar
  11. [D6] C. N. Delzell,Nonexistence of analytically varying solutions to Hilbert's 17th problem, Recent Advances in Real Algebraic Geometry and Quadratic Forms, Proc. RAGSQUAD Year, Berkeley 1990–1991 (W.B. Jacob, et al., eds.), Contemp. Math.155, Amer. Math. Soc. 1994, pp. 107–117Google Scholar
  12. [D7] C. N. Delzell, L. González-Bega, H. Lombardi,A continuous and rational solution to Hilbert's 17th problem and several cases of the Positivstellensatz, Compuational algebraic geometry (F. Eyssette, A. Galligo, eds.), Prog. Math.109, Birkhäuser (1993), 61–75Google Scholar
  13. [D8] R. A. Devore and G. G. Lorentz,Constructive approximation, Springer-Verlag, 1993Google Scholar
  14. [D9] L. E. Dickson,History of the theory of numbers, Vol. II. Chelsea, New York, 1966, Originally published by Carnegie Institute of Washington, 1920MATHGoogle Scholar
  15. [E1] W. J. Ellison,Waring's problem, Amer. Math. Monthly78 (1971), 10–36MATHCrossRefMathSciNetGoogle Scholar
  16. [H1] W. Habicht,Über die Zerlegung strikte definiter Formen in Quadrate. Comment. Math. Helv.12 (1940), 317–322MATHCrossRefMathSciNetGoogle Scholar
  17. [H2] G. H. Hardy, J. Littlewood and G. Pólya,Inequalities, 2nd ed., Cambridge University Press, 1967Google Scholar
  18. [H3] F. Hausdorff,Zur Hilbertschen Lösung des Waringschen Problems. Math. Ann.67 (1909), 301–305CrossRefMathSciNetGoogle Scholar
  19. [H4] D. Hilbert,Über die Darstellung definiter Formen als Summe von Formenquadraten Math. Ann.32 (1888), 342–350; see Ges. Abh. 2, 154–161, Springer, Berlin, 1933, reprinted by Chelsea, New York, 1981CrossRefMathSciNetGoogle Scholar
  20. [H5] D. Hilbert,Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem) Math. Ann.67 (1909), 281–300; see Ges. Abh. 1, 510–527, Springer, Berlin, 1932, reprinted by Chelsea, New York, 1981CrossRefMathSciNetGoogle Scholar
  21. [H6] E. W. Hobson,On a theorem in differentiation, and its application to spherical harmonics, Proc. of the London Math. Soc.24 (1892–1893), 55–67CrossRefGoogle Scholar
  22. [H7] E. W. Hobson,On a theorem in the differential calculus Mess. of Math.23 (1893–1894), 115–119Google Scholar
  23. [H8] E. W. Hobson,The theory of spherical and ellipsoidal harmonics, Cambridge University Press, 1931Google Scholar
  24. [P1] G. Pólya,Über positive Darstellung von Polynomen Vierteljschr. Naturforsch. Ges. Zürich73 (1928), 141–145; see Collected Papers, Vol. 2, pp. 309–313, MIT Press, 1974Google Scholar
  25. [R1] B. Reznick,Sums of even powers of real linear forms Mem. Amer. Math. Soc.96 (1992), no. 463Google Scholar
  26. [R2] B. Reznick,An inequality for products of polynomials Proc. Amer. Math. Soc.117 (1993), 1063–1073MATHCrossRefMathSciNetGoogle Scholar
  27. [R3] R. M. Robinson,Some definite polynomials which are not sums of squares in polynomials, Selected questions of algebra and logic (a collection dedicated to the memory of A. I. Mal'cev), Indat. “Nauka” Sibirsk. Otdel. Novosibirsk (1973), 264–282, Abstract in Notices Amer. Math. Soc. 16 (1969), p. 554Google Scholar
  28. [S1] J. Schmid,On totally positive units of real holomorphy rings Israel J. Math.85 (1994), 339–350MATHMathSciNetGoogle Scholar
  29. [S2] A. Strasburger,A generalization of the Bochner identity Exposition. Math.11 (1993), 153–157MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Bruce Reznick
    • 1
  1. 1.University of IllinoisUrbanaUSA

Personalised recommendations