# Sergei Yakovenko's blog: on Math and Teaching

## Friday, August 22, 2008

### Infinitesimal Hilbert 16th Problem

The number of limit cycles that can be born from periodic solutions of a polynomial Hamiltonian planar system $\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y),~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)$ by a small polynomial perturbation

$\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y)+\varepsilon P(x,y),~~~~~~~~\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y)-\varepsilon Q(x,y)$

not increasing the degree $n=\text{deg}H$, is explicitly bounded by a double exponent $2^{2^{\text{Poly}(n)}}$, where $\text{Poly}(n)$ is an explicit polynomial in $n$ of degree not exceeding 60 (fine tuning of the proof gives a better value around 5 or so, which hypothetically could be reduced to just 2). For hyperelliptic Hamiltonians of the form $H(x,y)=y^2+x^{n+1}+a_1 x^{n-1}+\cdots+a_{n-1}x+a_n$ the bound can be improved to $2^{2^{O(n)}}$ with an explicit constant in the term $O(n)$. This assertion constitutes an explicit constructive solution of the so called “Infinitesimal” Hilbert 16th Problem which first implicitly appeared in the works of Petrovskii and Landis in the 1950-s. Since mid-1960-s the problem was repeatedly formulated in many sources (starting with Arnold’s problems and as recently as in Ilyashenko’s 2008 list) as the natural step towards a still evasive solution of the complete Hilbert 16th Problem.

J’ai Nous (i.e., Gal Binyamini, Dmitry Novikov et moi-même) avons trouvé une merveilleuse démonstration de cette proposition, mais je ne peux l’écrire dans cette marge car elle est trop longue.”

La démonstration is indeed a bit too long to be reproduced here: the complete exposition is available on arXiv (50+ pages) and strongly uses another paper of 30+ pages which establishes non-uniform explicit double exponential upper bound on the number of isolated complex zeros of functions satisfying linear systems of Fuchsian differential equations, provided that all residue matrices have only real eigenvalues.
Our proof is based solely on the fact that Abelian integrals of polynomial 1-forms along cycles on complexified level curves of the Hamiltonian, satisfy an integrable system of regular Pfaffian differential equations defined over $\mathbb Q$ with quasiunipotent monodromy along all small loops.

Bookmark this page, as it will display the most up-to-date version of the text of both papers. Any comments, suggestions and spotted typos will be accepted with warmest gratitude.

1. Wow! That’s a mighty large number of them cycles!…

Comment by A student — Tuesday, August 26, 2008 @ 4:08

2. It seems that the precise quote left by maître Pierre on the margin of L’Arithmétique of Diophante, is not so easy to find. For instance, Wikipedia believes that the phrase was

…j’ai découvert une preuve réellement remarquable que cette marge trop étroite ne me permet pas de détailler.

According to the book Précis des Œvres Mathématiques (Par E. Brassinne), Toulouse, Imprimerie de Jean-Matthieu Douladourê, Rue Saint-Rome, No 41. 1855 (Reprinted: Éditions Jacques Gabay, © 1989), it reads as follows (p. 53),

…j’en ai assurément trouvé l’admirable démonstration. La marge trop exiguë ne la contiendrait pas…

By the way, this famous quote is not unique of its kind: on yet another occasion, Fermat writes (p. 130):

Je ne pense pas qu’on puisse donner sur les nombres un théorème plus beau et plus général. Je n’ai ni le loisir ni la convenance d’insérer la démonstration à la marge.

Comment by Sergei Yakovenko — Friday, August 29, 2008 @ 10:53

3. […] systems, Hilbert 16th problem, isomonodromic deformations Finally the two texts concerned with solution of the Infinitesimal Hilbert problem, are put into the polished form (including the publisher’s LaTeX style files). The new […]

Pingback by IH16 and friends: the final dash « Sergei Yakovenko’s Weblog — Wednesday, December 3, 2008 @ 11:03

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