Skip to main content
SpringerLink
Log in

Sommes de trois carrés de fractions en deux variables

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

This paper extends previous works by Cassels-Ellison-Pfister and Christie in producing different collections of positive polynomials in ℝ[X,Y] which are not sums of three squares of rational fractions. The results are essentially obtained in extending methods of Galois descent for positivity of the ℝ(X)-rank of special kinds of elliptic curves.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Hamb. Abh. 5, 100–115 (1927)

    Google Scholar 

  2. Cassels, J.W.S., Ellison, W.J., Pfister, A.: On sums of squares and on elliptic curves over function fields. J. Number Theory 3, 125–149 (1971)

    Article  Google Scholar 

  3. Christie, M.R.: Positive Definite Rational Functions of Two Variables Which Are Not the Sum of Three Squares. J. Number Theory 8, 224–232 (1976)

    Article  Google Scholar 

  4. Colliot-Thélène, J.-L.: The Noether-Lefschetz theorem and sums of 4 squares in the rational function field R(x,y). Compositio 86, 235–243 (1993)

    Google Scholar 

  5. Hilbert, D.: Über die Darstellung definiter Formen als Summen von Formenquadraten. Math. Ann. 32, 342–350, (1888) = Ges. Abh. 2, pp. 154–164

    Google Scholar 

  6. Huisman, J., Mahé, L.: Geometrical aspects of the level of curves. J. Algebra 239, 647–674 (2001)

    Article  Google Scholar 

  7. Knapp, A.: Elliptic curves. Math. Notes, Princeton U. Press, 1993

  8. Lam, T.Y.: The algebraic theory of quadratic forms. Mathematics Lecture Note Series. W. A. Benjamin, Inc., 1973

  9. S. Lang, Survey of diophantine geometry, Springer (1997)

  10. Macé, O.: Sommes de trois carrés en deux variables et représentation de bas degré pour le niveau des courbes réelles, Thèse de l’Université de Rennes 1, 2000, http://tel.ccsd.cnrs.fr/documents/archives0/00/00/62/39/index_fr. html

  11. Pfister, A.: Zur Darstellung definiter Funktionen als Summe von Quadraten. Invent. Math. 4, 229–237 (1967)

    MATH  Google Scholar 

  12. Silverman, J.H., Tate, J.: Rational points on elliptic curves. Undergraduate texts in mathematics, Springer-Verlag, 1992

Download references

Author information

Authors and Affiliations

  1. IRMAR, Campus de Beaulieu, 35042, Rennes-Cedex, France

    Olivier Macé & Louis Mahé

Authors
  1. Olivier Macé
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis Mahé
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Macé, O., Mahé, L. Sommes de trois carrés de fractions en deux variables. manuscripta math. 116, 421–447 (2005). https://doi.org/10.1007/s00229-004-0535-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-004-0535-0

Keywords

Search

Navigation