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Algebraic variety

: * This article is about algebraic varieties. For the term "a variety of algebras", and an explanation of the difference between a variety of algebras and an algebraic variety, see * [variety (universal algebra)] * . *

In [mathematics] , an ** algebraic variety ** is the [set of solutions] of a system of [polynomial] [equation] s. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) [algebraic geometry] .

The word "variety" is employed in the sense of a mathematical [manifold] , for which, in [Romance language] s, [cognate] s of the word "variety" are used.

Historically, the [fundamental theorem of algebra] established a link between algebra and geometry by saying that a [monic polynomial] in one variable over the [complex numbers] is determined by the set of its roots, which can be considered a geometric object. Building on this result, [Hilbert's Nullstellensatz] provides a fundamental correspondence between [ideals] of [polynomial ring] s and subsets of [affine space] . Using the Nullstellensatz and related results, we are able to capture the geometric notion of a [variety] in algebraic terms as well as bring geometry to bear on questions of [ring theory] .

** Formal definitions **

Algebraic varieties can be classed into four kinds: affine varieties, [quasi-affine varieties] , projective varieties, and [quasi-projective varieties] . There also exists the more general notion of an [abstract algebraic variety] .

** Affine varieties **

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