Hilbert's Sixteenth Problem

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, together with the other 22 problems.

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 polynomial vector fields of degree n and an investigation of their relative positions.

The first part of the problem is yet unsolved for n = 8. Therefore, this case is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. In the context of vector fields, an upper bound of the number of limit cycles is the first unsolved problem and is therefore what usually is meant by Hilbert's sixteenth problem in this context.