Hilbert's sixteenth problem
Encyclopedia
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 curve
s and surfaces (Problem der Topologie algebraischer Kurven und Flächen).
Actually the problem consists of two similar problems in different branches of mathematics:
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.
investigated algebraic curve
s and found that curves of degree n could have no more than
separate components
in the real projective plane. 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-curve
s.
Hilbert had investigated the M-curves of degree 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.
plane, that is a system of differential equations of the form:
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.
Hilbert continues:
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
at the Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
conference of the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in 1900, together with the other 22 problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
.
The original problem was posed as the Problem of the topology of algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s 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 curveAlgebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s 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.
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 HarnackCarl Gustav Axel Harnack
Carl Gustav Axel Harnack was a German mathematician who contributed to potential theory. Harnack's inequality applied to harmonic functions...
investigated algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s and found that curves of degree n could have no more than
separate components
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
in the real projective plane. 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-curve
Harnack's curve theorem
In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve...
s.
Hilbert had investigated the M-curves of degree 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 realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
plane, that is a system of differential equations of the form:
where both P and Q are real polynomials of degree n.
These polynomial vector fields were studied by Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, 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:Hilbert continues: