Cubic form

In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form.

In (Delone & Faddeev 1964), Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields. Their work was generalized in (Gan, Gross & Savin 2002, §4) to include all cubic rings,^[1][2] giving a discriminant-preserving bijection between orbits of a GL(2, Z)-action on the space of integral binary cubic forms and cubic rings up to isomorphism.

Examples

• Elliptic curve
• Fermat cubic
• Cubic 3-fold
• Koras–Russell cubic threefold
• Klein cubic threefold
• Segre cubic

Notes

1. ^ A cubic ring is a ring that is isomorphic to Z³ as a Z-module.
2. ^ In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.

References

• Delone, Boris; Faddeev, Dmitriĭ (1964) [1940, Translated from the Russian by Emma Lehmer and Sue Ann Walker], The theory of irrationalities of the third degree, Translations of Mathematical Monographs, 10, American Mathematical Society, MR0160744 
• Gan, Wee-Teck; Gross, Benedict; Savin, Gordan (2002), "Fourier coefficients of modular forms on G₂", Duke Mathematical Journal 115 (1): 105–169, doi:10.1215/S0012-7094-02-11514-2, MR1932327 
• Iskovskikh, V.A.; Popov, V.L. (2001), "Cubic form", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/c/c027260.htm 
• Iskovskikh, V.A.; Popov, V.L. (2001), "Cubic hypersurface", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/c/c027270.htm 
• Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library, 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR833513, http://books.google.com/books?id=W03vAAAAMAAJ 

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Homogeneous polynomials Multilinear algebra Algebraic geometry Algebraic varieties Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем написать курсовую Cubic Transportation Systems Cubic harmonic Look at other dictionaries: cubic form — kubinis pavidalas statusas T sritis fizika atitikmenys: angl. cubic form vok. kubische Form, f rus. кубическая форма, f pranc. forme cubique, f …   Fizikos terminų žodynas Cubic — Cubical redirects here. For the partition, see cubicle. Contents 1 Science and mathematics 2 Computing 3 Media …   Wikipedia Cubic reciprocity — is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word reciprocity comes from the form of the main theorem, which states that …   Wikipedia Cubic honeycomb — Type Regular honeycomb Family Hypercube honeycomb Schläfli symbol {4,3,4} Coxeter Dynkin diagram …   Wikipedia Cubic — Cu bic (k? b?k), Cubical Cu bic*al ( b? kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See {Cube}.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. [1913 Webster] 2. (Crystallog.) Isometric or… …   The Collaborative International Dictionary of English Cubic equation — Cubic Cu bic (k? b?k), Cubical Cu bic*al ( b? kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See {Cube}.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. [1913 Webster] 2. (Crystallog.) Isometric …   The Collaborative International Dictionary of English Cubic foot — Cubic Cu bic (k? b?k), Cubical Cu bic*al ( b? kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See {Cube}.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. [1913 Webster] 2. (Crystallog.) Isometric …   The Collaborative International Dictionary of English Cubic number — Cubic Cu bic (k? b?k), Cubical Cu bic*al ( b? kal), a. [L. cubicus, Gr. ?????: cf. F. cubique. See {Cube}.] 1. Having the form or properties of a cube; contained, or capable of being contained, in a cube. [1913 Webster] 2. (Crystallog.) Isometric …   The Collaborative International Dictionary of English Cubic function — This article is about cubic equations in one variable. For cubic equations in two variables, see elliptic curve. Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis where y = 0). It has 2 critical points. Here …   Wikipedia Cubic zirconia — A round brilliant cut cubic zirconia Cubic zirconia (or CZ) is the cubic crystalline form of zirconium dioxide (ZrO2). The synthesized material is hard, optically flawless and usually colorless, but may be made in a variety of different colors.… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”