algebraic geometry, branch of geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. , based on analytic geometry analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic.
..... Click the link for more information. , that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates Cartesian coordinates (kärtē`zhən)
..... Click the link for more information. ). In plane geometry an algebraic curve is the locus of all points satisfying the polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a₀xⁿ+a
..... Click the link for more information. 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, … x_n), where each n-tuple specifies a unique point and x₁, x₂, x₃, … x_n are members of a given field field, in algebra, set of elements (usually numbers) that may be combined under the operations of addition and multiplication so that it constitutes an additive group , the nonzero elements form a multiplicative group, and multiplication distributes over addition.
..... Click the link for more information. (e.g., the complex numbers). An algebraic hypersurface is the locus of all such points satisfying the polynomial equation f(x₁,x₂,x₃, … 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.

algebraic geometry

Study of geometric objects expressed as equations and represented by graphs in a given coordinate system. In contrast to Euclidean geometry, algebraic geometry represents geometric objects using algebraic equations (e.g., a circle of radius r is defined by x² + y² = r²). Objects so defined can then be analyzed for symmetries, intercepts, and other properties without having to refer to a graph.