# Analgebraic Geometry

It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus

analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.

It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.

For more details see the presentation prepared for the minicourse (or the shorter version of the talk) and the draft of the paper.

Any remarks and comments will be highly appreciated.

## Families of Fuchsian equations

A Fuchsian equation on $\mathbb C P^1$ with only quasiunipotent singularities admits an upper bound for the number of complex roots of its solutions, which depends on the equation, in particular, in the “magnitude” (slope), but also on the relative position of its singularities.

We are interested in finding conditions ensuring that this bound does not “explode”. The easiest way to formulate this is to consider parametric families of Fuchsian equations.

We will assume that the parametric family has the form

$L_\lambda u=0,\qquad L_\lambda=\sum_{k=0}^n p_k(t,\lambda)\partial^k,\quad p_k\in\mathbb C[t,\lambda]\qquad (*)$

with the coefficients $p_k$ polynomial in $t$ and rationally depending on the parameters $\lambda\in\mathbb P^m$ (one can consider them as homogeneous polynomials of the same degree on $\mathbb C^{m+1}$). For some values of $\lambda$ the operator $L_\lambda$ may degenerate (the leading coefficient vanishes identically, not excluding the degeneracy $L_\lambda\equiv0$). Such values, however, should constitute a proper algebraic subvariety $\Lambda\subset\mathbb P^m$.

Note that, because of the semicontinuity, it is sufficient to establish the global uniform bound for the number of isolated roots only for $\lambda\notin\Lambda$: complex roots cannot disappear in the blue sky…

We will impose the following qualitative conditions, imposed on the family (*).

1. Isomonodromy: when parameters change, the monodromy group remains “the same”.
2. Tameness (regularity): solutions $u_\lambda(t)$ of the equations grow at most polynomially when $\lambda\to\Lambda$.
3. Quasiunipotence: all singular points always have quasiunipotent monodromy.

The last condition is the “regularity” with respect to the parameters rather than with respect to the independent variable $t$. All conditions need to be accurately formulated, but one can give a simple example producing such families.

Consider a rational matrix-valued 1-form $\Omega$ on $\mathbb P^1\times\mathbb P^m$ with the polar locus $\varSigma\subset \mathbb P^1\times\mathbb P^m$ which is an algebraic divisor (singular hypersurface). Assume that the linear system $\mathrm dX=\Omega X$ is locally solvable and regular on $\varSigma$. Then for any fixed $\lambda$ the first row components of the (multivalued) matrix function $X(t,\lambda)$ satisfy a linear Fuchsian equation $L_\lambda u=0$ rationally depending on $\lambda$. This way we get the family of equations automatically satisfying the first two conditions above. It turns out that the third condition is sufficient to verify only for a generic equation of the family.

(Kashiwara theorem follows).

## Boundedness of the slope

In the arbitrary family (*) the slope $\angle L_\lambda$ is a semialgebraic function of the parameter $\lambda\notin\Lambda$, eventually undefined on the locus $\Lambda$ itself, and may well be unbounded.

However, in the isomonodromic regular family this is impossible.

(Grigoriev theorem follows)

Corollary: conformal pseudoinvariance of the slope.

## Wednesday, December 3, 2008

### IH16 and friends: the final dash

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), accepted (Oct. 2009).

## 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.

# 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.

Create a free website or blog at WordPress.com.