Definition:Affine Algebraic Variety

Definition
An affine algebraic variety is any set which is the intersection of the zero sets of polynomials.

More formally, a subset $$S \in X \ $$ is an affine algebraic variety if and only if there exists a set of polynomials $$f_i: X \to Y \ $$, where $$i \in I \ $$ is any index set, such that:
 * $$S = \bigcap_{i \in I} \left\{{y \in Y : f_i (y) = 0 \in Y}\right\} \ $$