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.^[1]

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.^[2]