Sergei Yakovenko’s Weblog

Wednesday, October 22, 2008

Coming out of the closet

Filed under: conference — Sergei Yakovenko @ 5:24
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A couple of weeks ago some two-thirds of the conspirators coworkers attended the workshop Equations aux dérivées partielles et théorie de Galois différentielle dit Malgrangefest in Luminy and delivered a talk on their work.

Slides from this talk are now available (static pdf, \approx2 Mb) for everybody to see.

Friday, August 22, 2008

Infinitesimal Hilbert 16th Problem

Filed under: links, research announcement — Sergei Yakovenko @ 11:17
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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.

Click for full size photo

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.

Thursday, May 29, 2008

Lecture 12 (May 29, 2008)

Filed under: lecture — Sergei Yakovenko @ 9:47
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Logarithmic singularities

  1. De Rham division lemma (and its generalization)
  2. Definition of a logarithmic pole: (scalar case). Residues.
  3. Logarithmic complex: principal lemma on Λ-closedness.
  4. Principal example: logarithmic complex for the normal crossings. Saito theorem.
  5. Closed logarithmic 1-forms: complete description. Darbouxian foliations.
  6. Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
  7. Schlesinger system: flat connexions with logarithmic poles along the diagonal.
  8. Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.

Recommended reading: the same notes, sect. 3-4.

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