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Algebraic Variety

Basic definition

The zero set of a collection of polynomials. An algebraic variety is one of the the fundamental objects in algebraic geometry.

Detailed definition

An algebraic variety is a generalization to n dimensions of algebraic curves. More technically, an algebraic variety is a reduced scheme of finite type over a field K. An algebraic variety V in R^n (or C^n) is defined as the set of points satisfying a system of polynomial equations f_i(x_1, ..., x_n) = 0 for i = 1, 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.

Related terms

Abelian variety | affine variety | Albanese variety | algebraic number theory | Brauer-Severi variety | category theory | Chow variety | commutative algebra | conic section | Picard variety | scheme | variety | Zariski topology

Educational grade level

graduate school level

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