Cubic form
Encyclopedia
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 , Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic field
s. Their work was generalized in to include all cubic rings, 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.
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
of degree 3, and a cubic hypersurface is the zero set of a cubic form.
In , Boris Delone and Dmitriĭ Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic field
Cubic field
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.-Definition:If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field...
s. Their work was generalized in to include all cubic rings, giving a discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
-preserving bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
between orbits of a GL(2, Z)-action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on the space of integral binary cubic forms and cubic rings up to isomorphism.
Examples
- Elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
- Fermat cubic
- Cubic 3-fold
- Koras–Russell cubic threefoldKoras–Russell cubic threefoldIn algebraic geometry, the Koras–Russell cubic threefolds are smooth affine contractible threefolds studied by that have a hyperbolic action of a one-dimensional torus with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action...
- Klein cubic threefoldKlein cubic threefoldIn algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equationV^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \, studied by .Its automorphism group is the group PSL2 of order 660...
- Segre cubicSegre cubicIn algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 dimensional projective space, studied by .The Segre cubic is the set of points of P5 satisfying the equations\displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = 0.Its intersection...