{{Short description|Homogeneous polynomial of degree 3}} 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 the case of a cubic form in three variables, the zero set is a cubic plane curve.
In {{harv|Delone|Faddeev|1964}}, Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields. Their work was generalized in {{harv|Gan|Gross|Savin|2002|loc=§4}} to include all cubic rings (a '''{{vanchor|cubic ring}}''' is a ring that is isomorphic to '''Z'''<sup>3</sup> as a '''Z'''-module),<ref>In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme.</ref> 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.
The classification of real cubic forms <math>a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3</math> is linked to the classification of umbilical points of surfaces. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus.<ref name=port>{{Citation|first=Ian R.|last=Porteous|title=Geometric Differentiation, For the Intelligence of Curves and Surfaces|isbn=978-0-521-00264-6|pages=350 |date=2001|publisher=Cambridge University Press| edition=2nd}}</ref>
==Examples== *Cubic plane curve *Elliptic curve *Fermat cubic *Cubic 3-fold *Koras–Russell cubic threefold *Klein cubic threefold *Segre cubic
==Notes== {{reflist}}
==References== *{{Citation | last1=Delone | first1=Boris | author-link=Boris Delone | last2=Faddeev | first2=Dmitriĭ | title=The theory of irrationalities of the third degree | publisher=American Mathematical Society | series=Translations of Mathematical Monographs | volume=10 | year=1964 | orig-year=1940, Translated from the Russian by Emma Lehmer and Sue Ann Walker | mr=0160744 }} *{{Citation | last1=Gan | first1=Wee-Teck | last2=Gross | first2=Benedict | author2-link=Benedict Gross | last3=Savin | first3=Gordan | title=Fourier coefficients of modular forms on ''G''<sub>2</sub> | year=2002 | journal=Duke Mathematical Journal | volume=115 | number=1 | pages=105–169 | doi=10.1215/S0012-7094-02-11514-2 | mr=1932327 | citeseerx=10.1.1.207.3266 }} *{{eom|id=c/c027260|first=V.A.|last= Iskovskikh|first2=V.L.|last2= Popov|author2-link=Vladimir L. Popov|title=Cubic form}} *{{eom|id=c/c027270|first=V.A.|last= Iskovskikh|first2=V.L.|last2= Popov|author2-link=Vladimir L. Popov|title=Cubic hypersurface}} *{{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link=Yuri Ivanovich Manin | title=Cubic forms | orig-year=1972 | url=https://books.google.com/books?id=W03vAAAAMAAJ | publisher=North-Holland | location=Amsterdam | edition=2nd | series=North-Holland Mathematical Library | isbn=978-0-444-87823-6 | mr=833513 | year=1986 | volume=4}}
* Category:Multilinear algebra Category:Algebraic geometry Category:Algebraic varieties
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