## Hilbert’s sixteenth problem

The International Congress of Mathematicians takes place every four years and is the most important mathematical conference. In 1900 the conference took place in Paris and David Hilbert gave a talk where he presented a list of 23 mathematical problems which he considered to be of special importance. This list has been a programme for the development of mathematics ever since. The individual problems are famous and are known by their numbers in the original list. Here I will write about the 16th problem. In fact the problem comes in two parts and I will say nothing about one part, which belongs to the domain of algebraic geometry. Instead I will concentrate exclusively on the other, which concerns dynamical systems in the plane. Consider two equations $\frac{dx}{dt}=P(x,y)$, $\frac{dy}{dt}=Q(x,y)$, where $P$ and $Q$ are polynomial. Roughly speaking, the problem is concerned with the question of how many periodic solutions this system has. The simple example $P(x,y)=y$, $Q(x,y)=-x$ shows that there can be infinitely many periodic solutions and that the precise question has to be a little different. A periodic solution is called a limit cycle if there is another solution which converges to the image of the first as $t\to\infty$. The real issue is how many limit cycles the system can have. The first question is whether for a given system the number $N$ of limit cycles is always finite. A second is whether an inequality of the form $N\le H(n)$ holds, where $H(n)$ depends only on the degree of the polynomials. $H(n)$ is called the Hilbert number. Linear systems have no limit cycles so that $H(1)=0$. Until recently it was not known whether $H(2)$ was finite. A third question is to obtain an explicit bound for $H(n)$. The first question was answered positively by Écalle (1992) and Ilyashenko (1991), independently.

The subject has a long and troubled history. Already before 1900 Poincaré was interested in this problem and gave a partial solution. In 1923 Dulac claimed to have proved that the answer to the first question was yes. In a paper in 1955 Petrovskii and Landis claimed to have proved that $H(2)=3$ and that $H(n)\le P_3(n)$ for a particular cubic polynomial $P_3$. Both claims were false. As shown by Ilyashenko in 1982 there was a gap in Dulac’s proof. After 60 years there was almost no progress on this problem. Écalle worked on it intensively for a long time. For this reason he produced few publications and almost lost his job. At this point I pause to give a personal story. Some years ago I was invited to give a talk in Bayrischzell in the context of the elite programme in mathematical physics of the LMU in Munich. The programme of such an event includes two invited talks, one from outside mathematical physics (which in this case was mine) and one in the area of mathematical physics (which in this case was on a topic from string theory). In the second talk the concept of resurgence, which was invented by Écalle in his work on Hilbert’s sixteenth problem, played a central role. I see this as a further proof of the universality of mathematics.

The basic idea of the argument of Dulac was as follows. If there are infinitely many limit cycles then we can expect that they will accumulate somewhere. A point where they accumulate will lie on a periodic solution, a homoclinic orbit or a heteroclinic cycle. Starting at a nearby point and following the solution until it is near to the original point leads to a Poincaré mapping of a transversal. In the case of a periodic limiting solution this mapping is analytic. If there are infinitely many limit cycles the fixed points of the Poincaré mapping accumulate. It follows that this mapping is equal to the identity, contradicting the limit cycle nature of the solutions concerned. Dulac wanted to use a similar argument in the other two cases. Unfortunately in that case the Poincaré mapping is not analytic. What we need is a class of functions which on the one hand is general enough to include the Poincaré mappings in this situation and on the on the other hand cannot have an accumulating set of fixed points without being equal to the identity. This is very difficult and is where Dulac made his mistake.

What about the second question? It has been known since 1980 that $P(2)\ge 4$. There is an example with three limit cycles which contain a common steady state and another which does not. It is problematic to find (or to portray) these with a computer since three are very small and one very large. More about the history can be found in an excellent article of Iyashenko (Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. 39, 301) which was the main source for what I have written. In March 2021 a preprint by Pablo Pedregal appeared (http://arxiv.org/pdf/2103.07193.pdf) where he claimed to have answered the second problem. I feel that some caution is necessary in accepting this claim. A first reason is the illustrious history of mistakes in this field. A second is that Pedregal himself produced a related preprint with Llibre in 2014 which seems to have contained a mistake. The new preprint uses techniques which are far away from those usually applied to dynamical systems. On the one hand this gives some plausibility that it might contain a really new idea. On the other hand it makes it relatively difficult for most people coming from dynamical systems (including myself) to check the arguments. Can anyone out there tell me more?