Swedish Student Partly Solves 16th Hilbert Problem

An anonymous reader writes "Swedish media report that 22-year-old Elin Oxenhielm, a student at Stockholm University, has solved a chunk of one of the major problems posed to 20th century mathematics, Hilbert's 16th problem. Norwegian Aftenposten has an English version of the reports."

Comments

Re:Wow he's good

[Flarenet]

I know you intended to be funny, but if you had loaded the article, you would have noticed that "he" is actually a "she" (and a fairly good looking she at that. :) But this is slashdot, and reading the article should not get in the way of a good joke!

Re:I remember

[CSharpMinor]

Click.

Legend: A student arrives late to math class and finds two problems written on the chalkboard. Assuming they're homework problems, he jots them down in his notebook and works on the equations over the next few days before turning his solutions in to the instructor. Several weeks later, the professor turns up at the student's door with the student's work written up for publication. The two problems were not a homework assignment; they were problems previously thought to be unsolvable which the instructor had used as examples in his lecture that day.

Origins: This has to be one of the ultimate academic wish-fulfillment fantasies: a student not only proves himself the smartest one in his class, but also bests his professor and every other scholar in his field of study.

As far as we know, this legend is based upon a true incident. (That is, a version of this legend that antedates a known true incident has not yet been discovered). George B. Dantzig, then a graduate student at the University of California, Berkeley, arrived late for a statistics class one day and found two problems written on the board. Not knowing they were examples of "unsolvable" statistics problems, he solved them as a homework assignment. Dantzig, who later became a staff mathematician at Stanford University, recounted his solving two "unsolvable" problems in a 1986 interview for College Mathematics Journal, and his solutions to the two problems can be found in the journal articles listed in the Sources section below.

problem description

[combinatorics]

Here's a description of the problem from
http://aleph0.clarku.edu/~djoyce/hilbert/toc.html
snip...A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space...

Can someone please post graphical, dumbed down representation of this problem so we can better understand it?

Re:It's funny that college kids....

[hurtstotouchfire]

Fermat had a full-time job as a respected jurist, and he was an extremely prolific mathematician.

However, Andrew Wiles, who solved Fermat's last theorem, spent seven years in his attic to do so.

I guess broad generalizations don't work so well, eh?

Context

[ixache]

I know that this is Slashdot and that around here the looks of a mathematician are more important than her work, but if anyone is interested, here are a few pointers to get to know more.

First, a short description of Hilbert's problems at Wolfram: Hilbert's Problems -- from MathWorld.

Then, a link to a text of Hilbert's original lecture in Paris in 1900.

Next, a quote of the 16-th problem as laid out by Hilbert. (Sorry, no fancy LaTeX here.)

16. Problem of the topology of algebraic curves and surfaces

The maximum number of closed and separate branches which a plane algebraic curve of the n-th order can have has been determined by Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the 6-th order, I have satisfied myself--by a complicated process, it is true--that of the eleven branches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maxi mum number of sheets which a surface of the 4-th order in three dimensional space can really have.

In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare's boundary cycles (cycles limites) for a differential equation of the first order and degree of the form dy/dx = Y/X where X and Y are rational integral functions of the n-th degree in x and y. Written homogeneously, this is X(y dz/dt - z dy/dt) + Y(z dx/dt - x dz/dt) + Z(x dy/dt - y dx/dt) = 0, where X, Y, and Z are rational integral homogeneous functions of the n-th degree in x, y, z, and the latter are to be determined as functions of the parameter t.

Finally, I'll quote the abstract from Miss Elin Oxenhielm's article On the second part of Hilbert's 16th problem :

Let k be an integer such that k is larger than or equal to zero, and let H be the Hilbert number. In this paper, we use the method of describing functions to prove that in the Lienard equation, the upper bound for H(2k+1) is k. By applying this method to any planar polynomial vector field, it is possible to completely solve the second part of Hilbert's 16th problem.

Author Keywords: Second part of Hilbert's 16th problem; Hilbert number; Lienard equation; Describing function; Limit cycle; Polynomial vector field

To get the full text of the article you must apparently have a subscription of pay a $30 fee. It is easily available if you follow the directions from the author's page as I did.

Hope this helps

Now allow me for a few comments: solving one of Hilbert's problem is a huge achievement, even it's only part of one. What is even more stricking is that it's coming from a woman. Don't get me wrong, I'm no sexist, quite the contrary. What I mean is that only very few women made it to be recorded in the history of the mathematical science at large: other than Hypatia of Alexandria; Maria Gaetana Agnesi; Sophie Germain; Ada Byron, Lady Lovelace; Sofia Kovalevskaya; Emmy Noether, not many names come to mind. It would be really nice to add another one, to begin, and then work up from there.

Xavier

(Attempted) explanation of Hilbert's 16th

[kevinatilusa]

Reading Hilbert's lecture and a couple other sources, here is what I THINK Hilbert is asking in his 16th problem. Take this with a grain of salt.

The first part of Hilbert's 16th problem asks about the relative number and position of the components of a curve of order n. In other words, if we look at the graph of an equation of nth degree in the plane, what might the graph look like? We can describe it fairly easily for small n.

If n=1, the first order equations are precisely the linear ones, so the curve always consists of a single unbounded component (the straight line).

If n=2, the general equation of the 2nd order is Ax^2+Bxy+Cy^2+Dx+Ey+F=0, also known as the equation of a conic section. Depending on the coefficients, the graph will be a point, a line, a parabola, two intersecting lines, an ellipse, or a hyperbola. Geometrically, all of the cases but the last are only a single component. Therefore an equation of the second order has at most two branches. When there are two branches, they both are unbounded.

The case n=3 is much more complicated, and involves the study of what are known as elliptic curves. Beyond that, it just gets worse.

What Hilbert wished to have investigated was the geometry of the branches in the case of the curves with the most branches. As it turns out, you can't just have any orientation. If n=6, for example, the greatest number of branches is 11, but if the curve has 11 branches then one of the branches will always lie completely inside another branch. The 16th problem asks what similar restrictions are required for other n, and what happens if we look in higher dimensions than the plane.

A related problem that Hilbert referred to in his problem was that of curves defined by differential equations instead of polynomials. Here the objects of interest are boundary cycles of first order (featuring no derivatives higher than the first) differential equations. I have not encountered this term before, but if I had to guess I would say a boundary cycle was a closed, limiting path of a function satisfying the differential equation (so, for example, a boundary cycle of the second-order differential equation given by gravitation would be a planet's orbit after it is sucked in the system). The same sort of question is asked: how could these cycles be placed relative to one another in the plane? It is this question that may have been answered by the student in the article.