With the introduction of Cartesian coordinates by R. Descartes and P. de Fermat in the seventeenth century, it was soon realized that equations in two variables generally define curves in the plane. Descartes had already made the distinction between curves that can be described by polynomial equations and those that cannot. He called the former geometrical and the latter mechanical curves. Nowadays, these are called algebraic and transcendental curves, respectively.

Algebraic geometry is the area of mathematics that studies the properties of sets (or loci) defined as the set of common zeros of a collection of polynomial equations on the coordinates of the points of some Cartesian coordinate system. Such sets are called algebraic sets or algebraic varieties. If they are one-dimensional, they are called algebraic curves, and if two-dimensional, algebraic surfaces. For example, a subset of the plane defined as the solution set of an...