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 revisions, already uploaded to ArXiv, differ from the initial submissions only by corrected typos, a few rearrangements aimed at improving the readability of the texts, and a couple of more references added. There is absolutely no need to read the new revision if you already have read the first one.
Mostly for the reasons of “internal convenience” the complete references are repoduced here:
- G. Binyamini and S. Yakovenko, Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems, posted as arXiv:0808.2950v2 [math.DS], 36 p.p., submitted to Ann. Inst. Fourier (Dec. 2008), accepted (February, 2009)
- G. Binyamini, D. Novikov and S. Yakovenko, On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem, posted as arXiv:0808.2952v2 [math.DS], 57 p.p., submitted to Inventiones Mathematicae (Nov. 2008).
Best wishes for the New 5769 (תשס”ט) Year to all readers of this blog!
שנה טובה
The number of limit cycles that can be born from periodic solutions of a polynomial Hamiltonian planar system 
by a small polynomial perturbation
not increasing the degree 
, is explicitly bounded by a double exponent 
, where 
is an explicit polynomial in 
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 
the bound can be improved to 
with an explicit constant in the term 
. 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 
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.
Actually, I forgot to tell the last time that the academic year is over. Congratulations to all the survivors who made it till the end. Hope you don’t regret.
Logarithmic singularities
- De Rham division lemma (and its generalization)
- Definition of a logarithmic pole: (scalar case). Residues.
- Logarithmic complex: principal lemma on Λ-closedness.
- Principal example: logarithmic complex for the normal crossings. Saito theorem.
- Closed logarithmic 1-forms: complete description. Darbouxian foliations.
- Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
- Schlesinger system: flat connexions with logarithmic poles along the diagonal.
- Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.
Recommended reading: the same notes, sect. 3-4.
Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification
- Pfaffian systems and their integrability
- From local to global solutions: monodromy
- Geometric language: covariant derivative and its curvature
- Meromorphic functions, meromorphic forms
- Example: multidimensional Euler system
- Regular singularities
- Flat connexions vs. isomonodromic deformations
Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.
No classes today, as 50% of the students are speaking on a conference elsewhere.
Stokes phenomenon for irregular singularities of linear systems
- Irregular singularities: total recall. Formal diagonalizability of non-resonant systems.
- Sectorial gauge equivalence: formal, holomorphic, asymptotic series.
- Separation rays. Sibuya theorem on sectorial normalization (statement).
- Sectorial authomorphisms. Rigidity of the normal form in large sectors.
- Stokes matrix cochain and Stokes matrix multipliers as complete invariants of holomorphic classification of irregular singularities.
- Stokes phenomenon. Realization theorem (Birkhoff). Generic divergence of the formal gauge normalizing transformations.
Recommended reading: Sections 20F-20I from the Book
Irregular singularities of linear systems
- One-dimensional case: complete classification.
- Polynomial “normal forms”: Birkhoff theorem and its “uselessness”.
- Local reducibility: similarities and differences with the regular (Fuchsian) case.
- Polynomial “normal form” for irreducible irregular singularity: Bolibruch theorem
- First steps of the “genuine” normal forms theory.
- Resonances.
- Formal diagonalizability of nonresonant systems
- Divergence of the normalizing transformations
Recommended reading: Section 20 from the Book
Notice
The next week there will be no classes for this reason. Expect the end of the story on May 1, 2008. In the meantime I wish to everybody חג פסח שמח and merry holidays.
Recommended reading: 
2 Mb) for everybody to see.
