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 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 Z³ as a Z-module),In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary scheme. 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 $a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3$ 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.{{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}}

Examples

Notes

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 G₂

| 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