Hilbert's sixteenth problem

Hilbert's sixteenth 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 algebraic curves of order n (and similarly for algebraic surfaces).
• The determination of the upper bound for the number of limit cycles in polynomial vector fields of order n and an investigation of their relative positions.

A request for an investigation is of course a rather open-ended problem, and it is thus doubtful, that those parts will ever be fully resolved. The search for the upper bound of the number of limit cycles in polynomial vector fields is therefore what usually is meant when talking about Hilbert's sixteenth problem.

The first part of Hilbert's 16th problem

In 1876 Harnack investigated algebraic curves and found that curves of order n could have no more than

<math> {n^2-3n+4 \over 2} <math>

separate 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 order 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 the 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:

<math> {dx \over dt}=P(x,y), \qquad {dy \over dt}=Q(x,y) <math>

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.

The original formulation of the problems

In his speech, Hilbert presented the problems as:

The upper bound of closed and separate branches of an algebraic curve of order n was decided by Harnack (Mathematische Annalen, 10); from this arises the further question as of the relative positions of the branches in the plane. As of the curves of order 6, I have - admittedly in a rather elaborate way - convinced myself that the 11 branches, that they can have according to Harnack, never all can be separate, rather there must exist one branch, which have another branch running in its interior and nine branches running in its exterior, or opposite. 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 order 4 in three-dimensional space can maximally have. (cf. Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886)

Hilbert continues:

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 and first degree on the form:

<math> {dy \over dx} = {Y \over X} <math>

where X, Y are integer, rational functions of n-th degree in resp. x, y, or written homogeneously:

<math>

X \left( y {dz \over dt} - z {dy \over dt} \right) + Y\left(z {dx \over dt} - x {dz \over dt} \right) + Z\left(x {dy \over dt} - y {dx \over dt} \right) = 0 <math> where X, Y, Z means integral, rational, homogenic functions of n-th degree in x, y, z and the latter are to be considered function of the parameter t.