Ternary quartic

In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.

Hilbert (1888) showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.

Salmon (1879) discussed the invariants of order up to about 15.

(Dolgachev 2012, 6.4) 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.

Table 2 from Noether's dissertation ( Noether 1908 ) on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u . The horizontal direction of the table lists the invariants with increasing grades in x , while the vertical direction lists them with increasing grades in u .