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.

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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.

Wednesday, December 19, 2007

“Auxiliary Lesson” שעור עזר) #9) Dec. 20, 2007

Dynamics generated by finitely generated subgroups of conformal germs

  1. Generic subgroups of \text{Diff}(\mathbb C^1,0) are non-solvable.
  2. Dynamics generated by several germs. Definition of a pseudogroup. Orbits of points.
  3. Periodicity of germs (finiteness of order) vs. periodicity of orbits. Cycles and limit cycles of pseudogroups.
  4. Convergence of elements in pseudogroups. Closure.
  5. Density of orbits. Linear subgroups. Abundance of limit cycles for generic (nonsolvable) subgroups of \text{Diff}(\mathbb C^1,0).
  6. Topological equivalence of subgroups and pseudogroups. Conjugacy of dense linear subgroups.
  7. Rigidity of nonsolvable subgroups: topological conjugacy implies holomorphic conjugacy.

Disclaimer… if somebody still needs it… 😦
Reading: Section 6 (second part) from the book, printing disabled.

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