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The Noether-Lefschetz theorem and sums of 4 squares in the rational function field R ( x , y )

J.-L. Colliot-Thélène

Compositio Mathematica (1993)

  • Volume: 86, Issue: 2, page 235-243
  • ISSN: 0010-437X

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Colliot-Thélène, J.-L.. "The Noether-Lefschetz theorem and sums of 4 squares in the rational function field $R(x, y)$." Compositio Mathematica 86.2 (1993): 235-243. <http://eudml.org/doc/90219>.

@article{Colliot1993,
author = {Colliot-Thélène, J.-L.},
journal = {Compositio Mathematica},
keywords = {rational functions; polynomials; sum of four squares; sum of squares; Noether-Lefschetz theorem},
language = {eng},
number = {2},
pages = {235-243},
publisher = {Kluwer Academic Publishers},
title = {The Noether-Lefschetz theorem and sums of 4 squares in the rational function field $R(x, y)$},
url = {http://eudml.org/doc/90219},
volume = {86},
year = {1993},
}

TY - JOUR
AU - Colliot-Thélène, J.-L.
TI - The Noether-Lefschetz theorem and sums of 4 squares in the rational function field $R(x, y)$
JO - Compositio Mathematica
PY - 1993
PB - Kluwer Academic Publishers
VL - 86
IS - 2
SP - 235
EP - 243
LA - eng
KW - rational functions; polynomials; sum of four squares; sum of squares; Noether-Lefschetz theorem
UR - http://eudml.org/doc/90219
ER -

References

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  1. 1 A. Buium: Sur le nombre de Picard des revêtements doubles des surfaces algébriques, C.R. Acad. Sc. Paris296 (1983) Série I, 361-364. Zbl0543.14017MR699163
  2. 2 J.W.S. Cassels: On the representation of rational functions as sums of squares, Acta Arithmetica9 (1964), 79-82. Zbl0131.25001MR162791
  3. 3 J W. S. Cassels, W. Ellison and A. Pfister: On sums of squares and on elliptic curves over function fields, J. Number Theory3 (1971), 125-149. Zbl0217.04302MR292781
  4. 4 M.R. Christie: Positive definite rational functions in two variables which are not the sum of three squares, J. Number Theory8 (1976), 224-232. Zbl0331.14017MR412162
  5. 5 J.-L. Colliot-Thélène:Real rational surfaces without a real point, Archiv der Mathematik.58 (1992) 392-396. Zbl0738.14023MR1152629
  6. 6 P. Deligne: Le théorème de Noether, in SGA 7 II, exp.XIX, SpringerL.N.M.340 (1973), 328-340. Zbl0269.14019
  7. 7 C. Delorme: Espaces projectifs anisotropes, Bull. Soc. Math. France103 (1975), 203-223. Zbl0314.14016MR404277
  8. 8 I. Dolgachev: Weighted projective varieties, in SpringerL.N.M.956 (1982), 34-71. Zbl0516.14014MR704986
  9. 9 L. Ein: An analogue of Max Noether's theorem, Duke Mathematical Journal52 (1985), 689-706. Zbl0589.14034MR808098
  10. 10 T. Ford: The Brauer group and ramified double covers of surfaces, preprint 1991. Zbl0779.13002MR1191982
  11. 11 P. Griffiths and J. Harris: On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann.271 (1985), 31-51. Zbl0552.14011MR779603
  12. 12 D. Hilbert: Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann.42 (1888), 342-350. Zbl20.0198.02MR1510517JFM20.0198.02
  13. 13 D. Hilbert: Ueber ternäre definite Formen, Acta Math.17 (1893), 169-197. JFM25.0319.01
  14. 14 T.-Y. Lam: The algebraic theory of quadratic forms, Benjamin/Cummings1973. Zbl0437.10006MR634798
  15. 15 E. Landau: Ueber die Darstellung definiter Funktionen durch Quadrate, Math. Ann.62 (1906), 272-285. MR1511376JFM37.0252.01
  16. 16 S. Lefschetz: On certain numerical invariants of algebraic varieties with application to Abelian varieties, Trans. Amer. Math. Soc.22 (1921), 327-482. Zbl48.0428.03MR1501178JFM48.0428.03
  17. 17 S. Mori: On a generalisation of complete intersections, J. of Math. of Kyoto University15 (1975), 619-646. Zbl0332.14019MR393054
  18. 18 J. Steenbrink: On the Picard group of certain smooth surfaces in weighted projective space, in Algebraic geometry, Proceedings, La Rabida, 1981, SpringerL.N.M.961 (1982), 302-313. Zbl0507.14025MR708341
  19. 19 T. Terasoma: Complete intersections with middle Picard number 1 defined over Q, Math. Z.189 (1985), 289-296. Zbl0579.14006MR779223

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