In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
The ternary cubic is one of the few cases of a form of degree greater than 2 in more than 2 variables whose ring of invariants was calculated explicitly in the 19th century.
The algebra of invariants of a ternary cubic under SL3(C) is a polynomial algebra generated by two invariants S and T of degrees 4 and 6, called Aronhold invariants. The invariants are rather complicated when written as polynomials in the coefficients of the ternary cubic, and are given explicitly in ( Sturmfels 1993, 4.4.7, 4.5.3)
The ring of covariants is given as follows. ( Dolgachev 2012, 3.4.3)
The identity covariant U of a ternary cubic has degree 1 and order 3.
The Hessian H is a covariant of ternary cubics of degree 3 and order 3.
There is a covariant G of ternary cubics of degree 8 and order 6 that vanishes on points x lying on the Salmon conic of the polar of x with respect to the curve and its Hessian curve.
The Brioschi covariant J is the Jacobian of U, G, and H of degree 12, order 9.
The algebra of covariants of a ternary cubic is generated over the ring of invariants by U, G, H, and J, with a relation that the square of J is a polynomial in the other generators.
( Dolgachev 2012, 3.4.3)
The Clebsch transfer of the discriminant of a binary cubic is a contravariant F of ternary cubics of degree 4 and class 6, giving the dual cubic of a cubic curve.
The Cayleyan P of a ternary cubic is a contravariant of degree 3 and class 3.
The quippian Q of a ternary cubic is a contravariant of degree 5 and class 3.
The Hermite contravariant Π is another contravariant of ternary cubics of degree 12 and class 9.
The ring of contravariants is generated over the ring of invariants by F, P, Q, and Π, with a relation that Π2 is a polynomial in the other generators.
Gordan (1869) and Cayley (1881) described the ring of concomitants, giving 34 generators.
The Clebsch transfer of the Hessian of a binary cubic is a concomitant of degree 2, order 2, and class 2.
The Clebsch transfer of the Jacobian of the identity covariant and the Hessian of a binary cubic is a concomitant of ternary cubics of degree 3, class 3, and order 3
- Cayley, Arthur (1881), "On the 34 Concomitants of the Ternary Cubic", American Journal of Mathematics, 4 (1): 1–15, doi: 10.2307/2369145, ISSN 0002-9327, JSTOR 2369145
- Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press, ISBN 978-1-107-01765-8
- Gordan, Paul (1869), "Ueber ternäre Formen dritten Grades", Mathematische Annalen, 1: 90–128, doi: 10.1007/bf01447388, ISSN 0025-5831, S2CID 123421707
- Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, CiteSeerX 10.1.1.39.2924, doi: 10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR 1255980
In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
The ternary cubic is one of the few cases of a form of degree greater than 2 in more than 2 variables whose ring of invariants was calculated explicitly in the 19th century.
The algebra of invariants of a ternary cubic under SL3(C) is a polynomial algebra generated by two invariants S and T of degrees 4 and 6, called Aronhold invariants. The invariants are rather complicated when written as polynomials in the coefficients of the ternary cubic, and are given explicitly in ( Sturmfels 1993, 4.4.7, 4.5.3)
The ring of covariants is given as follows. ( Dolgachev 2012, 3.4.3)
The identity covariant U of a ternary cubic has degree 1 and order 3.
The Hessian H is a covariant of ternary cubics of degree 3 and order 3.
There is a covariant G of ternary cubics of degree 8 and order 6 that vanishes on points x lying on the Salmon conic of the polar of x with respect to the curve and its Hessian curve.
The Brioschi covariant J is the Jacobian of U, G, and H of degree 12, order 9.
The algebra of covariants of a ternary cubic is generated over the ring of invariants by U, G, H, and J, with a relation that the square of J is a polynomial in the other generators.
( Dolgachev 2012, 3.4.3)
The Clebsch transfer of the discriminant of a binary cubic is a contravariant F of ternary cubics of degree 4 and class 6, giving the dual cubic of a cubic curve.
The Cayleyan P of a ternary cubic is a contravariant of degree 3 and class 3.
The quippian Q of a ternary cubic is a contravariant of degree 5 and class 3.
The Hermite contravariant Π is another contravariant of ternary cubics of degree 12 and class 9.
The ring of contravariants is generated over the ring of invariants by F, P, Q, and Π, with a relation that Π2 is a polynomial in the other generators.
Gordan (1869) and Cayley (1881) described the ring of concomitants, giving 34 generators.
The Clebsch transfer of the Hessian of a binary cubic is a concomitant of degree 2, order 2, and class 2.
The Clebsch transfer of the Jacobian of the identity covariant and the Hessian of a binary cubic is a concomitant of ternary cubics of degree 3, class 3, and order 3
- Cayley, Arthur (1881), "On the 34 Concomitants of the Ternary Cubic", American Journal of Mathematics, 4 (1): 1–15, doi: 10.2307/2369145, ISSN 0002-9327, JSTOR 2369145
- Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press, ISBN 978-1-107-01765-8
- Gordan, Paul (1869), "Ueber ternäre Formen dritten Grades", Mathematische Annalen, 1: 90–128, doi: 10.1007/bf01447388, ISSN 0025-5831, S2CID 123421707
- Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, CiteSeerX 10.1.1.39.2924, doi: 10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR 1255980