Definition:Affine Algebraic Variety

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

More formally, let $$F \ $$ be any field.

Then a subset $$S \in F^n \ $$ is an affine algebraic variety if and only if there exists a set of polynomials:
 * $$f_i: F^n \to F \ $$

where $$i \in I \ $$ is any index set such that:
 * $$S = \bigcap_{i \in I} \left\{{x \in F^n : f_i (x) = 0 \in Y}\right\} \ $$