The Spanish Royal Society for Mathematics published an explanation of Hilbert's sixteenth problem.
This statement is not obvious, since it is easy to construct smooth (C∞) vector fields in the plane with infinitely many concentric limit cycles.
[3] 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.
Evgenii Landis and Ivan Petrovsky claimed a solution in the 1950s, but it was shown wrong in the early 1960s.
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the corresponding investigation of the number, shape and position of the sheets of an algebraic surface in space – it is not yet even known, how many sheets a surface of degree 4 in three-dimensional space can maximally have.
Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886)Hilbert continues:[6] Following this purely algebraic problem I would like to raise a question that, it seems to me, can be attacked by the same method of continuous coefficient changing, and whose answer is of similar importance to the topology of the families of curves defined by differential equations – that is the question of the upper bound and position of the Poincaré boundary cycles (cycles limites) for a differential equation of first order of the form: where X, Y are integer, rational functions of nth degree in resp.