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 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 Clebsch transfer of the Hessian of a binary cubic is a concomitant of degree 2, order 2, and class 2.