Authors and affiliations
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
- Davenport, H., Cubic Forms in 16 Variables, Proceedings of the Royal Society A, Vol. 272, 1963, 285–303.
- Fujiwara, M., Hasse Principle in Algebraic Equations, Acta Arith., Vol. 22, 1973, 267–276.
- Heath-Brown, D. R., Cubic Forms in 14 Variables, Invent. Math., Vol. 170, 2007, No. 1,199–230.
- Hooley, C., On Nonary Cubic Forms, J. Für Die Reine Und Angewandte Mathematik,Vol. 386, 1988, 32–98.
- Matiyasevich, Y., Hilbert’s Tenth Problem, MIT Press, Massachusetts, 1993.
- 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.
- Schinzel, A., Hasse’s Principle for Systems of Ternary Quadratic Forms and for one Biquadratic form, Studia Mathematica, Vol. LXXVII, 1983, 103–109.
- Selmer, E., The Diophantine Equation ax3 + by3 + cz3=0, Acta Math., Vol. 85, 1951, No. 1, 203–362.
- Serre, J. P., A First Course in Arithmetic, Graduate Texts in Mathematics, Vol. 7, Berlin, New York: Springer–Verlag, 1973.
Cite this paperAPA
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.Chicago
Nguyen, Lan. “On the Local and Global Principle for System of Binary Rational Cubic Forms.” Notes on Number Theory and Discrete Mathematics 20, no. 5 (2014): 49-57.MLA
Nguyen, Lan. “On the Local and Global Principle for System of Binary Rational Cubic Forms.” Notes on Number Theory and Discrete Mathematics 20.5 (2014): 49-57. Print.