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manuscripta mathematica

, Volume 62, Issue 1, pp 21–32 | Cite as

On diagonal cubic surfaces

  • Andrew Bremner
Article

Abstract

We show that the affine surfacesx3+y3+cz3=c, cQ, in the casesc≠2,c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface.

However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.

Keywords

Number Theory Algebraic Geometry Topological Group Parametric Solution Arithmetic Genus 
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.

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References

  1. 1.
    LEHMER, D.H., On the Diophantine equationx 3+y 3+z 3=1, J. London Math. Soc., 31, 275–280, (1956).Google Scholar
  2. 2.
    SEGRE, B., On the rational solutions of homogeneous cubic equations in four variables, Math. Notae (Rosario), 11, 1–68, (1951).Google Scholar
  3. 3.
    SWINNERTON-DYER, H. P. F., Applications of algebraic geometry to number theory, 1969 Institute on Number Theory, Amer. Math. Soc. Proceedings XX, 1–52.Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Andrew Bremner
    • 1
  1. 1.Department of MathematicsArizona State UniversityTempe

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