# Real algebraic geometry

In

mathematics ,**real algebraic geometry**is the study ofreal number solutions toalgebraic equation s with real number coefficients.**Real plane curves**Since the

real number field is notalgebraically closed , the geometry of even aplane curve "C" in thereal projective plane is not a very easy topic. Assuming no singular points, the real points of "C" form a number of "ovals", in other words submanifolds that are topologically acircle . The real projective plane has afundamental group that is acyclic group with two elements. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane. Taking out out theline at infinity "L", any oval that stays in the finite part of theaffine plane will be contractible, and so represent the identity element of the fundamental group; the other type of oval must therefore intersect "L".There is still the question of how the various ovals are nested. This was the topic of a

Hilbert problem ,Hilbert's sixteenth problem . SeeHarnack's curve theorem for a classical result.**ee also***

Algebraic geometry **External links*** [

*http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/prob17.ps "The Role of Hilbert Problems in Real Algebraic Geometry" (PostScript)*]

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