In simplest terms, algebraic geometry is the study of the set of solutions of a system of polynomial equations. The main goal of real algebraic geometry is the study of real algebraic sets i.e. subsets of ℝ n defined by polynomial equations. By means of a simple example one can see some features which point up the difference between real and complex algebraic geometry. Let us consider the intersection of the straight line x = t, depending on the parameter t, with the cubic y 2 = x 3 - x. For t = -1,0,1 the straight line is tangent to the cubic. In the complex plane, when t is different from -1,0,1, the straight line always intersects the cubic in two points. In the real plane the situation is more intricate.
KeywordsAlgebraic Geometry Real Spectrum Euclidean Topology Real Algebraic Variety Real Algebraic Geometry
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