The original problem was posed as the Problem of the topology of algebraic curves and surfaces.
Actually the problem consists of two similar problems in different branches of mathematics:
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2 The second part of Hilbert's 16th problem 3 The original formulation of the problems 4 Attempts at solution |
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
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.
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 qualitive 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.
Hilbert continues:
Elin Oxenhielm, a 22 year old Swedish mathematics student has claimed to have recently solved part of the problem, however there has been great discussion as to whether her solution is correct or not, and related to the journal to which she had submitted her work.The second part of Hilbert's 16th problem
Here we are going to consider polynomial vector fields in the plane, that is a system of differential equations of the form:
where both P and Q are polynomial functions of degree n.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 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)
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 coefficientchanging, 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:
where X, Y are integer, rational functions of n-th degree in resp. x, y, or written homogeneously:
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.
Attempts at solution