mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a real plane curve is usually a real algebraic curve defined in the real projective plane.

Ovals

The field of real numbers is not algebraically closed, the geometry of even a plane curve ''C'' in the real projective plane. Assuming no singular points, the real points of ''C'' form a number of ''ovals'', in other words submanifolds that are topologically circles. The real projective plane has a

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

that is a cyclic 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 the line at infinity ''L'', any oval that stays in the finite part of the affine 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

Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topolog ...

. See Harnack's curve theorem for a classical result.

References

*{{Springer, id=P/p072800, title=Plane real algebraic curve Real algebraic geometry Algebraic curves

Ovals

See also

References