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| In [[mathematics]], a '''cubic form''' is a [[homogeneous polynomial]] of degree 3, and a '''cubic hypersurface''' is the [[zero set]] of a cubic form.
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| In {{harv|Delone|Faddeev|1964}}, [[Boris Delone]] and [[Dmitriĭ Faddeev]] showed that binary cubic forms with integer coefficients can be used to parametrize [[order (ring theory)|orders]] in [[cubic field]]s. Their work was generalized in {{harv|Gan|Gross|Savin|2002|loc=§4}} to include all cubic rings,<ref>A '''cubic ring''' is a [[ring (mathematics)|ring]] that is isomorphic to '''Z'''<sup>3</sup> as a [[Module (mathematics)|'''Z'''-module]].</ref><ref>In fact, [[Pierre Deligne]] pointed out that the correspondence works over an arbitrary [[Scheme (mathematics)|scheme]].</ref> giving a [[discriminant]]-preserving [[bijection]] between [[Orbit (group theory)|orbits]] of a GL(2, '''Z''')-[[Group action|action]] on the space of integral binary cubic forms and cubic rings up to [[isomorphism]].
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| 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 point]]s of surfaces. The [[equivalence class]]es of such cubics form a three dimensional [[real projective space]] and the subset of [[parabolic form]]s define a surface – the [[umbilic torus]] or [[umbilic bracelet]].<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|pp=350
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| |date=2001|publisher=Cambridge University Press| edition=2nd}}</ref>
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| ==Examples==
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| *[[Elliptic curve]]
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| *[[Fermat cubic]]
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| *[[Cubic 3-fold]]
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| *[[Koras–Russell cubic threefold]]
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| *[[Klein cubic threefold]]
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| *[[Segre cubic]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation
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| | last=Delone
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| | first=Boris
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| | author-link=Boris Delone
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| | last2=Faddeev
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| | first2=Dmitriĭ
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| | title=The theory of irrationalities of the third degree
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| | publisher=American Mathematical Society
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| | series=Translations of Mathematical Monographs
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| | volume=10
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| | year=1964
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| | origyear=1940, Translated from the Russian by Emma Lehmer and Sue Ann Walker
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| | mr=0160744
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| }}
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| *{{Citation
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| | last1=Gan
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| | first1=Wee-Teck
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| | last2=Gross
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| | first2=Benedict
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| | author2-link=Benedict Gross
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| | last3=Savin
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| | first3=Gordan
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| | title=Fourier coefficients of modular forms on ''G''<sub>2</sub>
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| | year=2002
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| | journal=Duke Mathematical Journal
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| | volume=115
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| | number=1
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| | pages=105–169
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| | doi=10.1215/S0012-7094-02-11514-2
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| | mr=1932327
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| }}
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| *{{eom|id=c/c027260|first=V.A.|last= Iskovskikh|first2=V.L.|last2= Popov|author2-link=Vladimir L. Popov}}
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| *{{eom|id=c/c027270|first=V.A.|last= Iskovskikh|first2=V.L.|last2= Popov|author2-link=Vladimir L. Popov|title=Cubic hypersurface}}
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| *{{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link=Yuri Ivanovich Manin | title=Cubic forms | origyear=1972 | url=http://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}}
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| [[Category:Homogeneous polynomials| ]]
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| [[Category:Multilinear algebra]]
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| [[Category:Algebraic geometry]]
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| [[Category:Algebraic varieties]]
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| {{Abstract-algebra-stub}}
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