Ternary cubic

In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.

Invariant theory

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 ring of invariants

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

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.

The ring of contravariants

(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.

The ring of concomitants

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

See also

• Ternary quartic
• Invariants of a binary form

References

• 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