Six legs on the table

The 16th problem in Hilbert's list consisted of two parts: the first dealt with planar algebraic equations, and the second with planar differential equations. Prof. Sergey Yakovenko, Head of the Department of Mathematics at the Weizmann Institute of Science, focused on a particular case of the 16th infinitesimal problem in his master's thesis under the guidance of Ilyashenko at Moscow University in the Soviet Union

On August 8, 1900, at the International Congress of Mathematicians held in Paris, David Hilbert, perhaps the most prominent mathematician of the late 19th and early 20th centuries, stood up and presented 10 open questions (out of a full list of 23 questions, some of which he chose not to answer) present at this event). "These," said Hilbert, "are the problems that should occupy mathematicians - and for which a solution will be found - in the 20th century." Most of the problems presented by Hilbert were indeed solved in the 20th century, but some of them are still open.

The 16th problem in Hilbert's list consisted of two parts: the first dealt with planar algebraic equations, and the second with planar differential equations. The second part of the problem was, in fact, formulated by Henri Poincare even before the year 1900, yet to this day there is no significant progress towards its solution. Despite the inherent difficulty of the problem, it can be formulated relatively simply: how many boundary circles can a plane differential equation of a given degree have?

This is perhaps the place to say that solutions of a plane differential equation are expressed as curved lines in the plane, which do not intersect themselves. When such a line repeats itself and "closes", it is called a circle. Usually, all the solutions close to the circle circle around it back and forth and their distance to it gets smaller and smaller with each round. In this case the circle is called a "boundary circle", since it is the limit of all its neighboring solutions. The importance of the boundary circles stems from the fact that they teach not only about themselves, but about all the solutions adjacent to them.

Various attempts to solve the problem over the years have ultimately been proven wrong. The most significant progress was made by the mathematicians Yuli Ilyishenko and Jean Acal, who proved (independently) that every differential equation in the plane has a finite number of boundary circles - but this work, which was a huge project, is still far from providing an answer to Hilbert's problem. When it became clear to the community of mathematicians that the problem as a whole resists all efforts to solve it, many mathematicians devoted their energy to "weakened" intermediate problems where, they hoped, they would be able to make progress.

One of the directions started with Ilyashenko's proposal to give up general differential equations at this stage and focus on special equations - the Hamiltonian equations. Equations of this type appear in physical formulations of mechanical systems, and they are known to be of special importance - since they have a perfect energy balance, and in particular, it turns out that they have no limit circuits at all.

But this is just the beginning. It turns out that even the smallest of the smallest changes usually breaks the energy balance of the equation, after which boundary circles are born, as if out of nowhere. Ilyashenko proposed to try and understand how many circuits can be formed in this way from the Hamiltonian system, or at least to determine an upper limit to their number. This problem has been studied under several names by different researchers, but by its most common name it is called "the 16th infinitesimal problem".

Prof. Serhiy Yakovenko, Head of the Department of Mathematics at the Weizmann Institute of Science, focused on a particular case of the 16th infinitesimal problem in his master's thesis supervised by Ilyashenko at Moscow University in the Soviet Union. "Since then," he says, "I return to this question again and again. It is, for me, a kind of lighthouse that constantly draws me to it." Years later, together with his student Dmitry Novikov, now a professor in the mathematics department at the Weizmann Institute of Science, Prof. Yakovenko managed to obtain some intermediate results. But the way to solving the 16th infinitesimal problem still remains blocked.

Another eight years passed, and those who peeked into Prof. Yakovenko's office encountered the silence of deep thought - and six legs resting carelessly on the table. The feet of Prof. Yakovenko, Dr. Novikov and the research student at the time, Gal Binyamini. "A war of minds in front of a blocking wall", Prof. Yakovenko describes it. "It was the most exciting and thrilling period in my professional life."

Dr. Binyamini: "One day while I was walking around the institute, as usual, I thought about the work of another student of Prof. Yakovenko, Dr. Alexei Grigoriev. Suddenly the thought occurred to me that the picture he drew could be 'twisted' or 'stretched' in different ways."

After each round and stretch it was possible to examine the ideas of Grigoriev, Yakovenko, Novikov and others from a new angle - and sometimes information was discovered that was difficult to see in the first place. The additional information obtained in this way allowed the three mathematicians from the Weizmann Institute of Science to determine an upper bound which constitutes a complete solution of the 16th infinitesimal problem.

"This," says Prof. Yakovenko, "is still very far from a complete solution of Hilbert's 16th problem. But the 16th infinitesimal problem also stood open for 50 years until we were able to find a solution for it by determining an upper limit. This work is one of the most significant advances that have taken place in the field in several decades."