Algebraic Variety

Algebraic Variety

In mathematics, an algebraic set is the set of solutions of a system of polynomial equations. Algebraic sets are sometimes also called algebraic varieties, but normally an algebraic variety is an irreducible algebraic set, i.e. one which is not the union of two other algebraic sets. Algebraic sets and algebraic varieties are the central objects of study in algebraic geometry.

The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold may not. In many languages, both varieties and manifolds are named by the same word.

Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex coefficients (an algebraic object) is determined by the set of its roots (a geometric object). Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specifity of algebraic geometry among the other subareas of geometry.