Real Algebraic Geometry
The real algebraic geometry is the study of the real points of the algebraic geometry.
The fact that the field of the reals number is an ordered field may not be occulted in such a study. For example, the curve of equation is a circle if, but does not have any real point if . It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the semi-algebraic sets, which are the solutions of systems of polynomial equations and polynomial inequalities. For example, a branch of the hyperbola of equation is not an algebraic variety, but is a semi-algebraic set defined by and or by and .
One of the challenging problems of real algebraic geometry is the unsolved Hilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.
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