# Sergei Yakovenko's blog: on Math and Teaching

## Riemann-Hilbert problem

The Riemann-Hilbert problem consists in “constructing a Fuchsian system with a prescribed monodromy”.

More precisely, let $M_1,M_2,\dots,M_d$ be nondegenerate matrices such that their product is an identical matrix, and $a_0,a_1,\dots, a_d\in\mathbb C$ are distinct points, such that the segments $[a_0,a_k]\subset\mathbb C,\ k=1,\dots,d$ are all disjoint except for the point $a_0$ itself.

The problem is to construct a linear system of equations

$\displaystyle \dot X=A(t)X,\quad A(t)=\sum_{k=1}^d \frac{A_k}{t-a_k},\quad \sum_{k=1}^d A_k=0$,

such that the monodromy operator along the path “$\gamma_k=$segment $[a_0,a_k]+$ small loop around $a_k+$segment $[a_k,a_0]$” is equal to $M_k$.

The modern strategy of solving this problem is surgery. One can easily construct a local solution, a differential system on a neighborhood $U_k$ of the segment $[a_0,a_k]$, which has the specified monodromy. The phase space of this system is the cylinder $U_k\times\mathbb C^n$, and without loss of generality one can assume that together the neighborhoods $U_k$ cover the whole Riemann sphere $\mathbb CP^1=\mathbb C\cup\{\infty\}$. Patching together these local solutions, one can construct a linear system with the specified monodromy, but it will be defined not on $\mathbb C P^1\times\mathbb C^n$, as required, but on a more general object, holomorphic vector bundle over $\mathbb C P^1$.

Description of different vector bundles is of an independent interest and is well known. It turns out (Birkhoff), that each holomorphic vector bundle in dimension $n$ is completely determined by a(n unordered) tuple of integer numbers $d_1,\dots,d_n\in\mathbb Z$, and the bundle is trivial if and only if $d_1=\cdots=d_n=0$.

However, the strategy of solving the Riemann-Hilbert problem by construction of the bundle and determining its holomorphic type is complicated by two facts:

1. Determination of the holomorphic type of a bundle is a transcendental problem;
2. The local realization of the monodromy is by no means unique: in the non-resonant case one can realize any matrix $M_k$ by an Euler system with the eigenvalues which can be arbitrarily shifted by integers; in the resonant case one should add to this freedom also non-Euler systems. This freedom can change the holomorphic type of the vector bundle in a very broad range.

It turns out that the fundamental role in solvability of the Riemann-Hilbert problem plays the (ir)reducibility of the linear group generated by the matrices $M_1,\dots,M_k$.

Theorem (Bolibruch, Kostov). If the group is irredicible, i.e., there is no invariant subspace in $\mathbb C^n$ common for all operators $M_k$, then one can choose the local realizations in such a way that the resulting bundle is trivial and thus yields solution to the Riemann-Hilbert problem.

The proof is achieved as follows: one constructs a possibly nontrivial bundle realizing the given monodromy, and then this bundle is brutally trivialized by a transformation that is only meromorphic at one of the singularities. The result will be a system with all but one singularities being Fuchsian, and the problem reduces to bringing to the Fuchsian form the last point (assumed to be at infinity) by transformations of the form $X\mapsto P(t)X$ with $P$ being a matrix polynomial with a constant nonzero determinant.  The group of such transformations is considerably more subtle, but ultimately the freedom in construction of the initial bundle can be used to guarantee that the last point is also “Fuchsianizible”.

All the way around, if the monodromy group is reducible, then there is an obstruction of the torsion type exists for trivializing the bundle. This obstruction was first discovered by A. Bolibruch, and its description can be found in the textbook by Yu. Ilyashenko and SY (sections 16G and 18).

## 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.

## 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.

# Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

1. Pfaffian systems and their integrability
2. From local to global solutions: monodromy
3. Geometric language: covariant derivative and its curvature
4. Meromorphic functions, meromorphic forms
5. Example: multidimensional Euler system
6. Regular singularities
7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

# Linear ordinary differential equations of order n

1. Construction of the Weyl algebra (noncommutative “differential polynomials of one independent variable”). Division with remainder, factorization, solutions.
2. Reconstruction of differential equations from their solutions. Riemann theorem.
3. Regular and Fuchsian operators. Complete local reducibility. Fuchs theorem (local regularity $\iff$ local Fuchs property) and its reformulations.

Recommended reading: Section 19 from the book (printing disabled)

# Bolibruch Impossibility Theorem

Revealing an obstruction for realization of a matrix group as the monodromy of a Fuchsian system on $\mathbb C P^1$.

1. Degree (Chern class) of a complex bundle vs. that of a subbundle. The total trace of residues of a meromorphic connexion.
2. Linear algebra: Monoblock operators and their invariant subspaces.
3. Local theory revisited: local invariant subbundles of a (resonant) Fuchsian singularity in the Poincaré–Dulac–Levelt normal form.
4. Bolibruch connexions on the trivial bundle: theorem on the spectra of residues.
5. Three Matrices $4\times 4$: the Bolibruch Counterexample.

Reading: Section 18E from the book (printing disabled).

Refresh your memory: Sections 16C16D (local theory), 17E-17I (degree of bundles)

## Tuesday, February 26, 2008

### Lecture 1 (Feb 27, 2008)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 9:55
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# Systems of Linear ODEs with complex time

1. Total recall: on differential equations in the complex domain (for the newcomers, if any) and foliations.
2. Linear systems: vector, matrix and Pfaffian form. Fyndamental solutions. Linearity of the transport maps.
3. Holomorphic (gauge) equivalence of linear systems. Monodromy group.
4. Linear systems with isolated singularities. Euler system and its properties.

Reading material: Section 15 from the Book (printing disabled)

## Topological properties of Abelian integrals

The second “learning in groups” meeting will be devoted to the study of the Gauss–Manin connexion in homology, which will ultimately result in a local representation of Abelian integrals as linear combinations of real powers and logarithms with analytic coefficients analytically depending on parameters.

This representation already suffices to produce local uniform bounds for the number of isolated zeros, as was explained on the previous Tuesday.

Recommended reading: Section 26 from the book (printing disabled), esp., subsections F and I-K.

Time and location: Tuesday Dec. 18, 2007, 14:00 (in place of the usual Geometry & Topology seminar time), Pekeris Room.

What it will be about:   😉

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