# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, December 30, 2014

### Final announcement

This is to inform the noble audience of the course that the main program of the course is completed. I will stay in Pisa for one more week (till January 8, 2015) and will be happy to discuss any subject (upon request).

Meanwhile one of the subjects discussed in this course was brought to a pre-final form: the manuscript

1. Shira Tanny, Sergei Yakovenko, On local Weyl equivalence of higher order Fucshian equations, arXiv:1412.7830,

was posted on arXiv and submitted to the Arnold Mathematical Journal, a new venue for publications molded in the spirit of  the late V. I. Arnol’d and his seminar.

Any criticism will be most appreciated. Congratulations modestly accepted.

Tanti auguri, carissimi! Buon anno, happy New Year, с наступающим Новым Годом, שנה (אזרחית) טובח, bonne année!

## 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).

## Noetherian chains

Computation of the (local intersectional) degree of a phase curve of a polynomial vector field, produced in Lecture 11, is based on the length of the ascending chain of polynomial ideals generated by consecutive derivations.

Let $D:\mathbb C[x_1,\dots,x_n]\to\mathbb C[x_1,\dots,x_n]$ be the Lie derivation of the algebra of polynomials along the vector field $v$. It increases the degrees by at most $d-1$. Let $p_0\in\mathbb C[x]$ be a seed polynomial of degree $\delta\in\mathbb N$ and consider the ascending chain of ideals

$I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq I_k\subseteq\cdots \subseteq\mathbb C[x],\qquad I_k=\left,$

where $p_k=Dp_{k-1},\ k=1,2,\dots$.  By Noetherianity, this chain must eventually stabilize at some step: $I_N=I_{N+1}=\cdots$. In addition to this chain of ideals, one can consider the associated descending chain of algebraic varieties

$\mathbb C^n\supseteq X_0\supseteq X_1\supseteq\cdots\supseteq X_k\supseteq\cdots, \qquad X_k=\{p_0(x)=\cdots=p_k(x)=0\}.$

This chain also stabilizes  no later than on the $N$th step, but may stabilize earlier.  The following properties of these chains can be verified by elementary arguments.

1. The chain of ideals is strictly ascending: If $I_N=I_{N+1}$, then all subsequent ideals in the chain coincide.
2. The chain of varieties may be nonstrictly ascending: e.g., $n=1,\ p_0(x)=x^m,\ D=\frac{\mathrm d}{\mathrm dx}$.
3. The length of the descending chain measures the maximal nontrivial order of contact between the trajectories of $v$ and the hypersurface $X_0=\{p_0=0\}$.

In general, the length of a strictly ascending chain of polynomial ideals generated by the sequence of polynomials of degrees not exceeding an explicit (growing) function of $k$, can be bounded by an algorithmically computable function. However, even in the simplest case where $\deg p_k\le \delta+k(d-1)$ (as above), this function turns out to be the Ackermann generalized exponential, a recursive but not primitively recursive function of $n,d,\delta\in\mathbb N$ which grows faster than any elementary (or primitive recursive) function. It is the algebraic origin of the sequence of polynomials, which allows to establish better results.

Example. Assume that $A:\mathbb C[x]\to\mathbb C[x]$ is an endomorphism of the ring of the polynomials, and instead of the iterations $p_k=Dp_{k-1}$ of the Lie derivation, we consider the sequence $p_k=Ap_{k-1}$. Then analogous chains can be constructed, yet their properties will be slightly different (in a sense, better). In particular, the chain of varieties becomes strictly descending and its length can be relatively simply bounded by simple function of $n,d,\delta$. If the growth rate of $\deg p_k$ is linear (as above), the bound will be double exponential in $n$. However,  in general the growth rate of iterates $A^k p_0$ is exponential, which leads to the bound given by a tower function (iterated exponent) of height $n=\dim x$.

The easiest way to estimate the length of varieties generated by consecutive derivations is based on the explicit Nullstellensatz. By this  theorem, for any polynomial $q\in\mathbb C[x]$ which vanishes on the variety $X\subseteq\mathbb C^n$ which is the zero locus of an ideal $I\subseteq\mathbb C[x]$ there exist a finite power $\rho$ such that $q^\rho\in X$. The number $\rho$ can be explicitly bounded from above (J. Kollar, 1988): if $I$ is generated by polynomials of degree no greater than $m$, then $\rho\leqslant m^n$. Having this bound, for each irreducible component of the variety $X_k$ which does not belong to the stable limit, one can predict, how many steps in can survive before being eliminated.  The resulting upper bound will be double exponential in $n$.

However, a better, more realistic and simple exponential in $n$ upper bound can be achieved by completely different argument.

Example. Assume that $n=2$ and we look at an isolated contact between a (nonsingular) trajectory of a vector field $v$ and an algebraic curve $X_0=X$ of degree $\delta$ at a point $a\in\mathbb C^2$. Consider the local analytic chart in which $v$ is parallel to the $y$-axis and the point $a$ is at the origin. If the curve $X$ has tangency of order $\mu$ with the vertical axis, then its projection on the $x$-axis is locally a ramified covering of order $\mu$. Consider a small bidisk neighborhood of the origin and apply a small analytic perturbation to $X$. The multiple tangency point will be scattered into several points of simple (quadratic) tangency, while the topological covering property will persist. Denote by $\nu$ the number of obtained simple tangencies: at each tangency exactly two leaves of the covering “collide”. Thus the total number of leaves $\mu$ cannot be greater than $2\nu$. The problem thus becomes to estimate $\nu$. However, the set of points of quadratic contact is algebraic: it is defined by the equations $X_1=\{p_0=0, p_1=0\}$ of degrees $\delta$ and $\delta+d-1$, so by the Bezout theorem the number of points does not exceed the product of these two numbers.

To generalize this argument for the multidimensional settings, one has to modify the topological part of the argument dealing with “exactly two leaves of the covering collide”. Instead of just one set $X_1$, one has to consider the sets $X_1,\dots, X_{n-1}$ (recall that $n$ is the dimension of the ambient space), and instead of counting points, one should consider their Euler characteristics. The corresponding combinatorics can be elegantly expressed by the “integration over the Euler characteristic” discovered by O. Viro (1988), while the bounds for the Euler characteristic of algebraic varieties can be bounded by virtue of the J. Milnor’s result (1964).

The result, due to A. Gabrielov and A. Khovanskii, is simple exponential (in $n$) bound, was achieved in 1998. However, for some problems in the analytic number theory (algebraic independence of transcendental numbers) it is important to have a more precise estimate of the maximal tangency order for $d, n$ fixed, but $\delta$ variable and growing to infinity. The most recent achievements in this direction are due to G. Binyamini [4], see below.

Besides, a different (and considerably more difficult) problem arises in the singular context, when one tries to estimate the order of contact of an algebraic hypersurface with a separatrix of a polynomial vector field, an invariant analytic curve (usually non-smooth) which contains a singular point of the vector field $v$.  Here again the most recent breakthroughs are due to Binyamini [5].

References (in addition to those mentioned earlier).

1. A. Gabrielov, A. KhovanksiiMultiplicity of a Noetherian intersection.  Geometry of differential equations, 119–130,
Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998.
2. O. ViroSome integral calculus based on Euler characteristic. Topology and geometry—Rohlin Seminar, 127–138,
Lecture Notes in Math., 1346, Springer, Berlin, 1988.
3. J. Milnor, On the Betti numbers of real varieties.  Proc. Amer. Math. Soc. 15 1964 275–280.
4. G. Binyamini, Multiplicity Estimates: a Morse-theoretic approach, arXiv:1406.1858 (2014).
5. G. Binyamini, Multiplicity estimates, analytic cycles and Newton polytopes, arXiv:1407.1183

## Monday, December 29, 2014

### Lecture 11 (Mon, Dec 15, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 4:43
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## Trajectories of polynomial vector fields

Definition
Let $\gamma:(-r,r)\to\mathbb R^n$ be a finite piece of a real analytic curve. Its intersection complexity (or intersection degree) is the maximal number of isolated intersections of $\gamma$ with real affine hyperplanes,

$\displaystyle \deg\gamma=\max_{\varPi\subset\mathbb R^n}\#\{t\in(-r,r):\gamma(t)\in\varPi\}<+\infty.$

The goal of this lecture is to explain the following result (D. Novikov, S.Y., 1999) which claims that a sufficiently small piece of a nonsingular trajectory of a polynomial vector field has a finite intersection degree bounded in terms of the dimension and the degree of the field.

More specifically, we consider the polynomial vector field associated with the system of polynomial differential equations

$\dot x_i=v_i(x_1,\dots,x_n),\qquad i=1,\dots,n,\quad v_i\in\mathbb R[x_1,\dots,x_n],\ \deg v_i\leqslant d$.

Denote its integral trajectory passing through an arbitrary point $a\in\mathbb R^n$ by $\gamma_a$.

Theorem
For any $a\in\mathbb R^n$ a sufficiently small piece of $\gamma_a$ has the intersection degree not exceeding $2^{2^{O(n^3 d)}}$.

References.

1. D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 563–609, MR1697373 (2001h:32054)
2. S. Yakovenko, Quantitative theory of ordinary dierential equations and the tangential Hilbert 16th problem, On finiteness in differential equations and Diophantine geometry, CRM Monogr. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109. MR2180125 (2006g:34062)

## 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.

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