Quartic Surface

In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.

More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form

where f is a polynomial of degree 4, such as f(x,y,z) = x4 + y4 + xyz + z2 − 1. This is a surface in affine space.

On the other hand, a projective quartic surface is a surface in projective space P3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f(x,y,z,w) = x4 + y4 + xyzw + z2w2 − w4.

If the base field in R or C the surface is said to be real or complex. If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.