# 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.

## Uniform bounds for parametric Fuchsian families

The previous lectures indicate how zeros of solutions can be counted for linear differential equations on the Riemann sphere. For an equation of the form

$u^{(n)} u+a_1(t)u^{(n-1)}u+\cdots+a_{n-1}(t)u'+a_n(t)u=0 ,\quad a_1,\dots, a_n\in\mathbb C(t)\qquad(*)$

one has to assume that:

1. The equation has only Fuchsian singularities at the poles of the coefficients $a_1,\dots,a_n$;
2. The monodromy of each singular point is quasiunipotent (i.e., all eigenvalues of the corresponding operator are on the unit circle);
3. The slope of the differential equation is known.

The slope is a badly formed and poorly computable number that characterizes the relative strength of the non-principal coefficients of the equation. It is defined as follows:

1. For a given affine chart $t\in\mathbb C$ on $\mathbb P^1$, multiply the equation (*) by the common denominator of the fractions for $a_k(t)$, reducing the corresponding operator to the form $b_0(t)\partial^n+b_1(t)\partial^{n-1}\cdots+b_n(t)$ with $b_0,\dots,b_n\in \mathbb C[t]$;
2. Define the affine slope as the $\max_{k=1,\dots,n}\frac{\|b_k\|}{\|b_0\|}$, where the norm of a polynomial $b(t)=\sum_j \beta_j t^j$ is the sum $\sum_j |\beta_j|$;
3. Define the conformal slope of an equation (*) as the supremum of the affine slopes of the corresponding operators over all affine charts on $\mathbb P^1$.
4. Claim. If the equation (*) is Fuchsian, then the conformal slope is finite.

The rationale behind the notion of the conformal slope of an equation is simple: it is assumed to be the sole parameter which allows to place an upper bound for the variation of arguments along “simple arcs” (say, circular arcs and line segments) which are away from the singular locus $\varSigma$ of the equation (*).

The dual notion is the conformal diameter of the singular locus. This is another badly computable but still controllable way to subdivide points of the singular locus into confluent groups that stay away from each other. The formal definition involves the sum of relative lengths of circular slits.

The claim (that is proved by similar arguments as the precious claim on boundedness of the conformal slope) is that a finite set points of the Riemann sphere $\mathbb P^1$ has conformal diameter bounded. Moreover, if $\varSigma\subseteq\mathbb P^m$ is an algebraic divisor of degree $d$ in the $m$-dimensional projective space, then the conformal diameter of any finite intersection
$\varSigma_\ell=\ell\cap\varSigma$ for any 1-dimensional line $\ell\subseteq\mathbb P^m$ is explicitly bounded in terms of $m,d$.

Together these results allow to prove the following general result.

Theorem (G. Binyamini, D. Novikov, S.Y.)

Consider a Pfaffian $n\times n$-system $\mathrm dX=\Omega X$ on the projective space $\mathbb P^m$ with the rational matrix 1-form of degree $d$. Assume that:

1. The system is integrable, $\mathrm d\Omega=\Omega\land\Omega$;
2. The system is regular, i.e., its solution matrix $X(t)$ grows at worst polynomially when $t$ tends to the polar locus
$\varSigma$ of the system;
3. The monodromy of the system along any small loop around $\varSigma$ is quasiunipotent.

Then the number of solutions of any solution is bounded in any triangle $T\subseteq\ell$ free from points of $late \varSigma$.

If in addition the system is defined over $\mathbb Q$ and has bitlength complexity $c$, then this number is explicitly bounded by a double exponential of the form $2^{c^{P(n,m,d)}}$, where $P(n,m,d)$ is an explicit polynomial of degree $\leqslant 60$ in these variables.

Remark. The quasiunipotence condition can be verified only for small loops around the principal (smooth) strata of $\varSigma$ by the Kashiwara theorem.

Reference

G. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert sixteenth problem, Inventiones Mathematicae 181 (2010), no. 2, 227-289, available here.

## 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).

## Wednesday, October 22, 2008

### Coming out of the closet

Filed under: conference — Sergei Yakovenko @ 5:24
Tags: , ,

A couple of weeks ago some twothirds 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, $\approx$2 Mb) for everybody to see.

## 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.

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