algebraic geometry, branch of geometry , based on analytic geometry , that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates ). In plane geometry an algebraic curve is the locus of all points satisfying the polynomial equation f ( x,y )=0; in three dimensions the polynomial equation f ( x,y,z )=0 defines an algebraic surface. In general, points in n -space are defined by ordered sequences of numbers ( x 1, x 2, x 3, … x n ), where each n -tuple specifies a unique point and x 1, x 2, x 3, … x n are members of a given field (e.g., the complex numbers). An algebraic hypersurface is the locus of all such points satisfying the polynomial equation f ( x 1, x 2, x 3, … x n )=0, whose coefficients are also chosen from the given field. The intersection of two or more algebraic hypersurfaces defines an algebraic set, or variety, a concept of particular importance in algebraic geometry.
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