Ternary quartic

In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
Hilbert's theorem.
 showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.
Invariant theory.
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . discussed the invariants of order up to about 15.
The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an
inflection bitangent. 
Catalecticant.
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.