# 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.

## Thursday, November 20, 2014

### Lecture 6 (Nov. 21, 2014)

Filed under: Analytic ODE course — Sergei Yakovenko @ 8:20
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## Zeros of solutions of linear equations

Nontrivial (i.e., not identically zero) solutions of linear ordinary differential equations obviously possess certain properties concerning their roots (points where these solutions vanish). The simplest, in a sense paradigmal property, is the following.

Prototheorem. Let $u$ be a nontrivial solution of a sufficiently regular linear ordinary differential equation $Lu=0$ of order $n>0$. Then $u$ cannot have a root of multiplicity greater or equal than $n-1$.

Here by regularity we mean the condition that the operator $L=\partial^n+a_1(t)\partial^{n-1}+\cdots+a_{n-1}(t)\partial+a_n(t)$ has coefficients smooth enough to guarantee that any solution $u(t)$ near any point $a$ in the domain of its definition is uniquely determined by the initial conditions $u(a),u'(s),\dots,u^{(n-1)}(a)$.

Indeed, if $u$ has a root of multiplicity $n$, that is, all first $n-1$ derivatives of $u$ at $a$ vanish, then $u^{(n)}(a)=0$ by virtue of the equation and hence the by the uniqueness $u(t)$ must be identically zero.

In particular, solutions of first order equation $u'+a_1(t)u=0$ are nonvanishing, solutions of any second order equation $u''+a_1(t)u'+a_2(t)u=0$ may have only simple roots etc.

Theorem (de la Vallee Poussin, 1929). Assume that the coefficients of the LODE

$u^{(n)}+a_1(t)u^{(n-1)}+\cdots+a_{n-1}(t)u'+a_n(t)u=0,\qquad t\in[0,\ell],\qquad (\dag)$

are explicitly bounded,  $|a_k(t)|\leqslant A_k\in\mathbb R_+,\ \forall t\in[0,\ell],\ k=1,\dots,n$.

Assume that the bounds are small relative to the length of the interval, i.e.,

$\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<1.\qquad (*)$

Then any nontrivial solution of the equation has no more than $n-1$ isolated roots on $[0,\ell]$ .

## Novikov’s counterexample

What about linear systems of the first order?

Consider the system $\dot x=A(t)x$ with $x=(x_1,\dots,x_n)\in \mathbb R^n$ and the norm $\|A(t)\|$ explicitly bounded on $[0,\ell]$. Consider all possible linear combinations $u=\sum_k c_k x_k(t),\ c\in\mathbb R^n$. Can one expect a uniform upper bound for the number of roots of all combinations?

Let $a(t)$ be a polynomial having many zeros on $[0,t]$. Consider the $2\times 2$-system of the form

$\displaystyle \dot x_1=a(t)x_1,\qquad \dot x_2=(\dot a+ a^2)x_1.$

The first equation defines a nonvanishing function $x_1(t)$, the second equation – its derivative which vanishes at all roots of $a(t)$.

By replacing $a(t)$ by $\varepsilon a(t)$ one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment $[0,\ell]$ (or even any open complex neighborhood of this real segment). Thus no matter how small are the coefficients, the second component will have the specified number of isolated roots.

## Complexification

What about complex valued versions? There is no Rolle theorem for them.

I will describe three possible replacements, Kim’s theorem (1963), nearest in the spirit, and two versions of the argument principle.

Theorem (W. Kim)
Assume that an analytic LODE

$u^{(n)}+a_1(z)u^{(n-1)}+\cdots+a_{n-1}(z)u'+a_n(z)u=0,\qquad z\in D\subseteq\mathbb C$

is defined in a convex compact subset $D$ of diameter $\ell$ and the condition (*) holds. Then this equation is disconjugate in $D$: any solution has at most $n-1$ isolated roots.

This result follows from the interpolation inequality of the following type: if $u(z)$ is a function holomorphic in $D$ and has $n$ isolated roots there, then $\|u\|_D\leqslant \frac{\ell^n}{n!}\|u^{(n)}\|$ (the maximum modulus norm is assumed).

Consider the equation $(\dag)$ on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves $u:[0,\ell]\to\mathbb C$ which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin $0\in\mathbb C$, which is defined as $R(u)=|\mathrm{Arg}~u(\ell)-\mathrm{Arg}~u(0)|$ for any continuous choice of the argument.

Theorem. Assume that

$\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<\frac12.$

Then rotation of any nontrivial solution $u$ is explicitly bounded: $R(u)<\pi (n+1)$.

If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle $D$, then application of this result to the sides of the triangle yields an explicit upper bound for the number of isolated roots of analytic solutions inside the triangle.

Reference

S. Yakovenko, On functions and curves defined by differential equations, §2.

## Wednesday, November 12, 2014

### Lecture 4 (Nov. 14, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 5:47
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## Algebraic theory of linear ordinary differential operators

• Differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs of functions of one variable $t\in(\mathbb C^1,0)$ + derivation $\partial =\frac{\mathrm d}{\mathrm dt}$ produce noncommutative polynomials $\Bbbk[\partial]$: a polynomial $L=\sum_{j=0}^n a_j\partial ^{n-j}$ acts on $\Bbbk$ in a natural way.
• The equation $Lu=0$ only exceptionally rarely has a solution in $\Bbbk$, but one can always construct a differential extension of $\Bbbk$ which will contain solutions of this equation.
• Analytically solutions of the equation form a tuple of functions $(u_1,\dots,u_n)$ analytic and multivalued in a punctured neighborhood of the origin. The multivaluedness is very special: the linear span remains the same after the analytic continuation, hence there exists a matrix $M\in\mathrm{GL}(n,\mathbb C)$ such that $\Delta (u_1,\dots,u_n)=(u_1,\dots, u_n)\cdot M$.
• Instead of $\partial$, any other derivation can  be used, in particular, the Euler derivation $\epsilon=t\partial$.
• Example. Equations with constant coefficients have the form $L=\sum c_j \partial^{n-j}$ with constant coefficients $c_j\in\mathbb C$. Such an operator can always be factorized into commuting factors, $L=c_0\,\prod_{\lambda_i\in\mathbb C} (\partial-\lambda_i)^{\nu_i}$ with $\sum\nu_i=n=\deg L$. A fundamental system of solutions consists of quasipolynomials $q_{ik}(t)=\mathrm e^{\lambda_i}t^k$, $0\leqslant k < \nu_i$. In a similar way the Euler operator has the form $L=\sum c_j\epsilon^{n-j}$ and its solutions are functions $u_{ik}=t^{\lambda_i}\ln^k t$, $k=0,1,\dots,\nu_i-1$ (look at the model equation $\epsilon^\nu u=0$).
• Weyl equivalence of of two operators. Two operators $L,M\in\Bbbk[\partial]$ of the same order are called Weyl equivalent, if there exist an operator $H\in\Bbbk[\partial]$ which maps any solution $u$ of the equation $Lu=0$ to a solution  $v=Hu$ of the equation $Mv=0$ isomorphically (i.e., no solution is mapped to zero).
The above definition means that the composition $MH$ vanishes on all solutions of $Lu=0$, hence must be divisible by $L$: $MH=KL$ for some $K\in\Bbbk[\partial]$.Note that the operator represented by each side of the above equality, is a non-commutative analog of the least common multiple of mutually prime polynomials $H,L$: it is divisible by both $L$ and $H$.
• Theorem. The Weyl equivalence is indeed an equivalence relationship: it is reflexive, symmetric and transitive.
The only thing that needs to be proved is the symmetry. Since $H, L$ are mutually prime, there exist two operators $U,V\in\Bbbk[\partial]$ such that $UL+VH=1$,  hence $LUL+LVH=L$. This identity means that $LVH$ is simultaneously divisible by $L$ and by $H$ (immediately). Hence $LVH$ is divisible by their least common multiple $KL=MH$: there exists an operator $W\in\Bbbk[\partial]$ such that $LVH=W\cdot MH=WMH$. But since the algebra $\Bbbk[\partial]$ is without zero divisors, the right factor $H$ can be cancelled, implying $LV=WM$, which means that $V$ maps solutions of $Mv=0$ into those of $Lu=0$.
• Different flavors of Weyl equivalence: regular (nonsingular) requiring $H, K$ be nonsingular or arbitrary.
• Theorem. Any nonsingular operator $L=\partial^n+\sum_1^n a_j \partial^j$ with holomorphic coefficients $a_j\in\mathscr O(\mathbb C,0)$, is regular Weyl equivalent to the operator $M=\partial^n$.
This result is analogous to the rectification theorem reducing any nonsingular system $\mathrm dX=\Omega X$ to $\mathrm dX=0$.
• Theorem. Any Fuchsian operator is Weyl equivalent to an Euler operator.
This is similar to the meromorphic classification of tame systems. The conjugacy $H$ may be non-Fuchsian.
• Missing part: a genuine analog of holomorphic classification of Fuchsian systems.

## Poincare-Dulac-Fuchs classification of Fuchsian operators

Instead of representing operators as non-commutative polynomials in $\partial$ or in $\epsilon$, one can represent them as non-commutative (formal) Taylor series of the form $L=\sum_{k\geqslant 0}t^k p_k(\epsilon)$ with the coefficients $p_k\in\mathbb C[\epsilon]$ from the commutative algebra of univariate polynomials, but not commuting with the “main variable” $t$.

Such an operator is Fuchsian of order $n$, if and only if $\deg p_k\leqslant n$ for all $k=1,2,\dots$, and $\deg p_0=n$. The polynomial $p_0$ is the “eulerization” of $L$, and the series can be considered as a noncommutative perturbation of the Euler operator $L_0=p_0\in\mathbb C[\epsilon]$.

Definition. The operator $L=p_0+tp_1+\cdots$ is non-resonant, if no two roots of $p_0$ differ by a nonzero integer, $\lambda_i-\lambda_j\notin\mathbb Z^*$.

Theorem. A non-resonant Fuchsian operator is Weyl equivalent to its Euler part with the conjugacy $H$ being a Fuchsian operator, $H=h_0+th_1+\cdots$, $\deg h_0\leqslant n-1,\ \gcd(p_0,h_0)=1$.

## In search of the general theory (to be continued)

References.

The classical paper by Ø. Ore (1932) in which the theory of non-commutative polynomials was established, and the draft of the paper by Shira Tanny and S.Y., based on Shira’s M.Sc. thesis (Weizmann Institute of Science, 2014).

## Local theory of Fuchsian systems (cont.)

• Resonant normal form.
Definition. A meromorphic Fuchsian singularity $\dot X=t^{-1}(A_0+tA_1+\cdots+t^k A_k+\cdots)X$, $A_0=\mathrm{diag}(\lambda_1,\dots,\lambda_n)+\mathrm N$, is in the (Poincare-Dulac) normal form, if for all $k=1,2,\dots$, the identities $t^\Lambda A_k t^{-\Lambda}=t^k A_k$ hold.
• Theorem. Any Fuchsian system is holomorphically gauge equivalent to a system in the normal form.
• Integrability of the normal form: let $I=\mathrm N+A_1+\cdots +A_k+\cdots$ (in fact, the sum is finite). Then the solution is given by the (non-commutative) product $X(t)=t^\Lambda t^I$. The monodromy is the (commutative) product, $M=\mathrm e^{2\pi \mathrm i \Lambda}\mathrm e^{2\pi\mathrm i I}$.

References: [IY], section 16.

## Linear high order homogeneous differential equations

• Differential operators as noncommutative polynomials in the variable $\partial=\frac {\mathrm d}{\mathrm dt}$ with coefficients in a differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs at the origin.
• Composition and factorization.
• Reduction of a linear equation $Lu=0$ to a system of linear first order equations and back. Singular and nonsingular equations.
• Euler derivation $\epsilon=t\partial$ and Fuchsian equations (“nonsingular with respect to $\epsilon$“).
• Division with remainder, greatest common divisor of two operators, divisibility and common solutions of two equations.
• Sauvage theorem. Tame equations are Fuchsian.

References: [IY], Section 19.

### Lecture 2 (Nov 7, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 6:04
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## Local theory of Fuchsian singular points

• Monodromy and holonomy.
• Growth of multivalued solutions.
• Tame singularities.
• Principal example: the Euler system $\dot X=\frac At X$, $A\in\mathfrak{gl}(n,\mathbb C)$. Solution:
$X(t)=t^A=\exp (A \ln t)$, monodromy $\Delta X(t) =X(t)M$, $M=\mathrm e^{2\pi\mathrm i A}$.
• Fuchsian condition.
• Gauge classification of linear systems, $A(t)\Longleftrightarrow \dot H(t)H^{-1}(t)+H(t)A(t)H^{-1}(t)$.
• Meromorphic gauge classification of tame (regular) systems.
• Holomorphic gauge classification of Fuchsian singularities: $A(t)=\frac 1t(A_0+tA_1+t^2A_2+\cdots)$,
$A_0=\Lambda+\mathrm N$, $\Lambda=\mathrm{diag}(\lambda_1,\dots,\lambda_n)$, $\mathrm N^n=0$.
• Resonances (integer differences between eigenvalues of $A_0$.
• Holomorphic Eulerization of non-resonant Fuchsian singularities.

Reference: [IY], section 16.

## Monday, November 3, 2014

### Lecture 1 (Nov. 3, 2014)

Filed under: Analytic ODE course — Sergei Yakovenko @ 6:34
Tags: , ,

The first lecture was introductory, containing the motivation for the forthcoming subjects.

The world of (real or complex) algebraic sets is tame: any question on the topological complexity admits an algorithmic solution and explicitly bounded answer.In particular, any algebraic set which consists of finitely many isolated points, admits an explicit bound for the number of these points by the product of degrees of equations defining this set (Bézout theorem). All the way around, equations involving nonalgebraic solutions to even simplest algebraic differential equations (sine/cosine), may define infinite sets (integer numbers). We will try to find out how the algebraic universe can be enlarged to include transcendental objects which still admit explicit bounds on their complexity.

It turns out that periods, integrals of rational forms over algebraic cycles, do possess such constructive finiteness, although this is far from easy to see. This finiteness is characteristic for solutions of rational Pfaffian systems with moderate singularities and special monodromy group.

## Part 1: General linear systems.

A linear system locally lives on a cylinder, the product of a (complex) linear space $\mathbb C^n$ and an open base $U\subset \mathbb C^k$. If $\Omega=\bigl(\omega_{ij}\bigr)$ is an $n\times n$-matrix of holomorphic 1-forms on the base $U$, then a linear system defined by this matrix 1-form, is a matrix differential equation $\mathrm dX=\Omega\cdot X$, whose solution is a holomorphically invertible matrix function $X=X(t)$, $t\in U$. If the base is one-dimensional, then $\Omega=A(t)\,\mathrm dt$ with a holomorphic matrix function $A(t)$, and the linear system takes the familiar shape $\dot X(t)=A(t)X(t)$ [IY, sect. 15]

A necessary and sufficient condition for a local existence of solution is vanishing of the curvature, which amounts to the  matrix identity $\mathrm d\Omega=\Omega\land\Omega$ (the right hand side is the matrix 2-form with the entries $\sum_{\ell=1}^k \omega_{i\ell}\land\omega_{\ell j}$, $i,j=1,\dots,k$).  See [NY, sect. 1].

Solution of a linear system is defined modulo a right multplicative constant matrix factor: $\mathrm d(XC)=\Omega XC$ for any $C\in\mathrm{GL}(n,\mathbb C)$, and any other solution has such form. Using this observation, any piecewise curve $latex\gamma$ on the base can be covered by small neighborhoods $U_\alpha$ with local solutions $X_\alpha$ in these neighborhoods, which agree on the pairwise intersections $U_\alpha\cap U_\beta$. If this was not the case for the initial choice of local solutions, this can be always achieved by suitably twisting them (replacing by $X_\alpha C_\alpha$ so that $X_\alpha C_\alpha=X_\beta C_\beta$ on the intersections). This explains how solutions can be continued analytically along any simple curve, yet after continuation along a closed path $\gamma$ the solution may acquire a non-trivial monodromy factor.

# 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.

# 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.

# Stokes phenomenon for irregular singularities of linear systems

1. Irregular singularities: total recall. Formal diagonalizability of non-resonant systems.
2. Sectorial gauge equivalence: formal, holomorphic, asymptotic series.
3. Separation rays. Sibuya theorem on sectorial normalization (statement).
4. Sectorial authomorphisms. Rigidity of the normal form in large sectors.
5. Stokes matrix cochain and Stokes matrix multipliers as complete invariants of holomorphic classification of irregular singularities.
6. Stokes phenomenon. Realization theorem (Birkhoff). Generic divergence of the formal gauge normalizing transformations.

Recommended reading: Sections 20F-20I from the Book

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