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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 4 + y 4 + x y z + z 2 − 1 {\displaystyle f=x^{4}+y^{4}+xyz+z^{2}-1}. This is a surface in affine space A.
On the other hand, a projective quartic surface is a surface in projective space P of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f = x 4 + y 4 + x y z w + z 2 w 2 − w 4 {\displaystyle f=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}}.