viewed as K-valued functions on by evaluating at the points in , i.e. by choosing values in K for each xi. For each set S of polynomials in , define the zero-locus Z(S) to be the set of points in on which the functions in S simultaneously vanish, that is to say A subset V of is called an affine algebraic set if V = Z(S) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is also called an affine variety. (Some authors use the phrase affine variety to refer to any affine algebraic set, irreducible or not.) Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets. This topology is called the Zariski topology. Given a subset V of , we define I(V) to be the ideal of all polynomial functions vanishing on V: For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal. Projective varieties and quasi-projective varieties Let be an algebraically closed field and let be the projective n-space over . Let in be a homogeneous polynomial of degree d. It is not well-defined to evaluate on points in in homogeneous coordinates. However, because is homogeneous, meaning that , it does make sense to ask whether vanishes at a point . For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in on which the functions in S vanish: A subset V of is called a projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety. Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of , let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal. A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine