On the local and global principle for system of binary rational cubic forms

Lan Nguyen
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 5, Pages 49—57
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Authors and affiliations

Lan Nguyen
Department of Mathematics, University of Wisconsin-Parkside
Ann Arbor, MI 48109, United States


It is known that any binary rational cubic form satisfies the Hasse principle. The next natural question to ask is whether this still holds for a system of binary rational cubic forms. However, there seems to be no known result on this topic. In our paper we show, by establishing an explicit equivalence between a rational cubic form and an intersection of quadric surfaces, that any system of finitely many binary rational cubic forms satisfies the Hasse principle.


  • Hasse principle
  • Cubic plane curve
  • Cubic form
  • Quadratic form
  • System of cubic forms
  • System of binary quadratic forms
  • Selmer curve
  • Finite Basis theorem

AMS Classification

  • 11XXX


  1. Davenport, H., Cubic Forms in 16 Variables, Proceedings of the Royal Society A, Vol. 272, 1963, 285–303.
  2. Fujiwara, M., Hasse Principle in Algebraic Equations, Acta Arith., Vol. 22, 1973, 267–276.
  3. Heath-Brown, D. R., Cubic Forms in 14 Variables, Invent. Math., Vol. 170, 2007, No. 1,199–230.
  4. Hooley, C., On Nonary Cubic Forms, J. Für Die Reine Und Angewandte Mathematik,Vol. 386, 1988, 32–98.
  5. Matiyasevich, Y., Hilbert’s Tenth Problem, MIT Press, Massachusetts, 1993.
  6. Mordell, L., On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees, Proc. Cambridge Phil. Soc., Vol. 21, 1922–1923, 179–192.
  7. Schinzel, A., Hasse’s Principle for Systems of Ternary Quadratic Forms and for one Biquadratic form, Studia Mathematica, Vol. LXXVII, 1983, 103–109.
  8. Selmer, E., The Diophantine Equation ax3 + by3 + cz3=0, Acta Math., Vol. 85, 1951, No. 1, 203–362.
  9. Serre, J. P., A First Course in Arithmetic, Graduate Texts in Mathematics, Vol. 7, Berlin, New York: Springer–Verlag, 1973.

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Cite this paper

Nguyen, L. (2014). On the local and global principle for system of binary rational cubic forms. Notes on Number Theory and Discrete Mathematics, 20(5), 49-57.

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