An investigation of the relative positions of the branches of real algebraic curves of degree n.
The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions.
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.
The first part of Hilbert's 16th problem
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.
The second part of Hilbert's 16th problem
Here we are going to consider polynomial vector fields in the real plane, that is a system of differential equations of the form: where both P and Q are real polynomials of degree n. These polynomial vector fields were studied by Poincaré, who had the idea of abandoning the search for finding exact solutions to the system, and instead attempted to study the qualitative features of the collection of all possible solutions. Among many important discoveries, he found that the limit sets of such solutions need not be a stationary point, but could rather be a periodic solution. Such solutions are called limit cycles. The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree n and, similar to the first part, investigate their relative positions.
Results
It was shown in 1991/1992 by Yulii Ilyashenko and Jean Écalle that every polynomial vector field in the plane has only finitely many limit cycles. This statement is not obvious, since it is easy to construct smooth vector fields in the plane with infinitely many concentric limit cycles. The question whether there exists a finite upper bound H for the number of limit cycles of planar polynomial vector fields of degree n remains unsolved for any n > 1. Evgenii Landis and Ivan Petrovsky claimed a solution in the 1950s, but it was shown wrong in the early 1960s. Quadratic plane vector fields with four limit cycles are known. An example of numerical visualization of four limit cycles in a quadratic plane vector field can be found in. In general, the difficulties in estimating the number of limit cycles by numerical integration are due to the nested limit cycles with very narrow regions of attraction, which are hidden attractors, and semi-stable limit cycles.
The original formulation of the problems
In his speech, Hilbert presented the problems as: Hilbert continues:
