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 topology of algebraic curves and surfaces (Problem der Topologie algebraischer Kurven und Flächen).

Actually the problem consists of two similar problems in different branches of mathematics:

The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upper bound for the number of limit cycles is known for any n > 1, and this is what usually is meant by Hilbert's sixteenth problem in the field of dynamical systems.

In 1876 Harnack investigated algebraic curves in the real projective plane and found that curves of degree n could have no more than

separate connected components. Furthermore, he showed how to construct curves that attained that upper bound, and thus that it was the best possible bound. Curves with that number of components are called M-curves.

Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.

Furthermore, he requested a generalization of Harnack's Theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.

Here we are going to consider polynomial vector fields in the real plane, that is a system of differential equations of the form:

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