In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables.
The zeros form a cubic surface in 3-dimensional projective space.
Salmon also gave an explicit formula for the discriminant as a polynomial in the generators, though Edge (1980) pointed out that the formula has a widely copied misprint in it.
A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cube roots of unity.
The union of the 5 planes where these 5 linear forms vanish is called the Sylvester pentahedron.