Алгебралық қисықтар мен беттердің топологиясы: Гильберттің 16-шы мәселесі

On topology of algebraic curves and surfaces
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:
An investigation of the relative positions of the branches of real algebraic curves of degree n (and similarly for algebraic surfaces). 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 Spanish Royal Society for Mathematics published an explanation of Hilbert's sixteenth problem.

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 (a 1923 article by Henri Dulac claiming a proof of this statement had been shown to contain a gap in 1981). This statement is not obvious, since it is easy to construct smooth (C∞) vector fields in the plane with infinitely many concentric limit cycles. The question whether there exists a finite upper bound H(n) for the number of limit cycles of planar polynomial vector fields of degree n remains unsolved for any n > 1. (H(1) = 0 since linear vector fields do not have limit cycles.) 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. 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.