An algebraic variety is a generalization to dimensions of algebraic curves.
More technically, an algebraic variety is a reduced scheme
of finite type over a field . An algebraic variety in (or ) is defined as the set of points satisfying a system of
polynomial equations for , 2, .... According to the Hilbert
basis theorem, a finite number of equations suffices.

A variety is the set of common zeros to a collection of polynomials. In classical algebraic geometry, the polynomials have complex
numbers for coefficients. Because of the fundamental
theorem of algebra, such polynomials always have zeros. For example,