Real plane curve

Inmathematics,areal plane curveis usually a realalgebraic curvedefined in thereal projective plane.

Ovals

The field ofreal numbersis notalgebraically closed,the geometry of even a plane curveCin thereal projective plane.Assuming nosingular points,the real points ofCform a number ofovals,in other words submanifolds that are topologicallycircles.The real projective plane has afundamental groupthat is acyclic 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 theline at infinityL,any oval that stays in the finite part of theaffine planewill be contractible, and so represent the identity element of the fundamental group; the other type of oval must therefore intersectL.

There is still the question of how the various ovals are nested. This was the topic ofHilbert's sixteenth problem.SeeHarnack's curve theoremfor a classical result.

See also

• Real algebraic geometry
• Ragsdale conjecture

References

• "Plane real algebraic curve",Encyclopedia of Mathematics,EMS Press,2001 [1994]