Harnack's curve theorem

For any algebraic curve of degree m in the real projective plane, the number of components c is bounded by The maximum number is one more than the maximum genus of a curve of degree m, attained when the curve is nonsingular.

Moreover, any number of components in this range of possible values can be attained.

A curve which attains the maximum number of real components is called an M-curve (from "maximum") – for example, an elliptic curve with two components, such as

or the Trott curve, a quartic with four components, are examples of M-curves.

This theorem formed the background to Hilbert's sixteenth problem.

The elliptic curve (smooth degree 3) on the left is an M-curve, as it has the maximum (2) components, while the curve on the right has only 1 component.
The Trott curve , shown here with 7 of its bitangents , is a quartic (degree 4) M-curve, attaining the maximum (4) components for a curve of that degree.