ternary cubic

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 {{harv|Sturmfels|1993|loc=4.4.7, 4.5.3}}

The ring of covariants is given as follows. {{harv|Dolgachev|2012|loc=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.

{{harv|Dolgachev|2012|loc=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.

{{harvtxt|Gordan|1869}} and {{harvtxt|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

• Ternary quartic
• Invariants of a binary form

• {{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=On the 34 Concomitants of the Ternary Cubic | jstor=2369145 | year=1881 | journal=American Journal of Mathematics | issn=0002-9327 | volume=4 | issue=1 | pages=1–15 | doi=10.2307/2369145| url=https://rcin.org.pl/dlibra/publication/edition/152691/content | url-access=subscription }}
• {{Citation | authorlink=Igor Dolgachev | last1=Dolgachev | first1=Igor V. | title=Classical Algebraic Geometry: a modern view | url=http://www.math.lsa.umich.edu/~idolga/CAG.pdf | publisher=Cambridge University Press | isbn=978-1-107-01765-8 | year=2012}}
• {{Citation | last1=Gordan | first1=Paul | title=Ueber ternäre Formen dritten Grades | year=1869 | journal=Mathematische Annalen | issn=0025-5831 | volume=1 | pages=90–128 | doi=10.1007/bf01447388| s2cid=123421707 | url=https://zenodo.org/record/1428246 }}
• {{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in invariant theory | publisher=Springer-Verlag | location=Berlin, New York | series=Texts and Monographs in Symbolic Computation | isbn=978-3-211-82445-0 | doi=10.1007/978-3-211-77417-5 | year=1993 | mr=1255980| citeseerx=10.1.1.39.2924 }}

Category:Invariant theory