algebraic geometry

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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 (x1,x2,x3, … xn), where each n-tuple specifies a unique point and x1, x2, x3, … xn 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(x1,x2,x3, … xn)=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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coordinate geometry (algebraic geometry) Branch of mathematics combining the methods of pure geometry with those of algebra. Any geometrical point can be given an algebraical value by relating it to coordinates, marked off from a frame of reference. Thus, if a point is marked on a square grid so that it is x1 squares along the x axis and y1 squares along the y axis, it has the coordinates (x1, y1). Polar coordinates can also be used. It was first introduced in the 17th century by René; Descartes. See also Cartesian coordinates