# Real plane curve

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

## Ovals

Since the real number field is not algebraically closed, the geometry of even a plane curve *C* in the real 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 circles. The real projective plane has a fundamental group 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. See Harnack's curve theorem for a classical result.

## See also

## References

- Hazewinkel, Michiel, ed. (2001), "Plane real algebraic curve",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4