Real algebraic geometry
mathematics, real algebraic geometry is the study of real numbersolutions to algebraic equations with real number coefficients.
Real plane curves
real number fieldis not algebraically closed, the geometry of even a plane curve"C" in the real projective planeis 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 a circle. The real projective plane has a fundamental groupthat is a cyclic groupwith 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 the line at infinity"L", any oval that stays in the finite part of the affine planewill 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. See Harnack's curve theoremfor a classical result.
* [http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/prob17.ps "The Role of Hilbert Problems in Real Algebraic Geometry" (PostScript)]
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