# 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.

## 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!

## Oscillatory behavior of Fuchsian equations

### Semilocal theory

Consider a holomorphic linear equation in the unit disk $0<|t|\le 1$, having a unique Fuchsian singularity at the origin $t=0$. Such an equation can be always reduced to the form $Lu=0,\ L=\epsilon^n+a_1(t)\epsilon^{n-1}+\cdots+a_n(t)$, with holomorphic bounded coefficients $a_1,\dots,a_n\in\mathscr O(D)$, $D=\{|t|\leqslant 1\}$, $|a_k(t)|\leqslant A$.

The previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial solution $u$ of the equation $Lu=0$ along the boundary of the unit disk $\partial D$: $\left.\mathrm{Var\,Arg\,}u(t)\right|_{t=1}^{t=\mathrm e^{2\pi \mathrm i}}\leqslant V_L=C\cdot n(A+1)$ for some universal constant $C$.

If the solution itself is holomorphic (e.g., in the case of apparent singularities), such bound would imply (by virtue of the argument principle) a bound for the number of zeros of $u$ in $D$. Unfortunately, solutions are usually ramified and the argument principle does not work. Denote by $\mathbf M$ the monodromy operator along the boundary.

##### Definition

The Fuchsian point is called quasiunipotent, if all eigenvalues $\mu_1,\dots,\mu_n$ of the matrix $\mathbf M$ have modulus one, $|\mu_k|=1$.

##### Theorem 1

The number of isolated roots of any solution of the equation $Lu=0$ in the Riemann domain $\Pi=\{0<|t|\leqslant 1,\ |\mathrm{Arg\,}t\le 2\pi\}$ having real coefficients $a_k(\mathbb R)\subseteq\mathbb R,\ k=1,\dots,n$ and a single quasiunipotent singularity at the origin does not exceed $(2n+1)(2V_L+1)$, where $V_L=Cn(A+1)$ is the parameter bounding the magnitude of coefficients of $L$.

The proof is based on a version of the flavor of the Rolle theorem for the “difference operators” $\mathbf P_\mu=\mu^{-1}\mathbf M-\mu\mathbf M^{-1}$ for any unit $\mu$ such that $\mu^{-1}=\bar\mu$:

$\#\{t\in\Pi:\ u(t)=0\}\leqslant \#\{t\in\Pi:\ \bigl(\mathbf P_\mu u\bigr)(t)=0\}+2V_L.$

A version of the Cayley-Hamilton theorem asserts that the (commutative) composition $\mathbf P=\prod_{\mu}\mathbf P_\mu$ over all eigenvalues of the monodromy operator (counted with their multiplicities) vanishes on all solutions of the real Fuchsian equation.

### Global theory

A linear ordinary differential equatuib with rational coefficients from $\Bbbk=\mathbb C(t)$ can always be transformed to the form

$Lu=0,\qquad p_0(t)\partial^n+p_1(t)\partial^{n-1}+\cdots+p_n(t),\qquad p_0,\dots,p_n\in\mathbb C[t].\qquad (*)$

It may depend on additional parameters $\lambda=(\lambda_1,\dots,\lambda_r)\in\mathbb C^r$: if this dependence is rational, then we may assume that the coefficients of the operator are polynomials from $\mathbb C[t,\lambda]$. The new feature then will be appearance of singular perturbations: for some values of the parameters $\lambda=\lambda_*$ the leading coefficient $p_0(~\cdot~,\lambda_*)$ may vanish identically in $t$, meaning that the order of the corresponding equation drops down to a smaller value. Such phenomenon is known to cause numerous troubles of analytic nature.

Changing the independent variable $\tau=1/t$ allows to investigate the nature of singularity at the infinite point $t=\infty\in\mathbb C P^1$. The equation is called Fuchsian, if it is Fuchsian at each its singular point on the Riemann sphere $\mathbb C P^1=\mathbb C\cup\{\infty\}$.

Assume that infinity is non-singular (this can always be achieved by a Mobius transformation of the independent variable $t$). Then a Fuchsian equation with the singular locus $Z=\{z_1,\dots,z_m\}\subset\mathbb C$ can always be transformed to the form $Mu=0$, where $M$ is the operator

$M= E^n+q_1(t)E^{n-1}+\cdots+q_{n-1}(t)E+q_n(t),$

$E=E_Z=(t-z_1)\cdots(t-z_m)\partial$

(nonsingularity at infinity implies certain bounds on the degrees of the polynomials $p_k\in\mathbb C[t]$). However, the coefficients of this form depend in the rational way not only on the coefficients of the original equation (*), but also on the location of the points $\{z_1, \dots,z_m\}$.

#### Definition.

The slope of this operator (*) is defined as the maximum

$\displaystyle\angle L=\max_{k=1,\dots,n}\frac{\|p_k\|}{\|p_0\|}$

where the norm of a polynomial $p(t)=\sum_0^r c_j t^j\in\mathbb C[t]$ is the sum $\|p\|=\sum_j |c_j|$.

Simple inequalities:

1. Any polynomial $p(t)$ of known degree $d=\deg p$ and norm $M=\|p\|$ admits an explicit upper bound for $|p(t)|$ on any disk $\{|t|\leqslant R\}$: $|p(t)| \leqslant MR^d$ for $R>1$.
2. A polynomial of unit norm $\|p\|=1$ admits a lower bound for $|p(t)|$ for points distant from its zero locus $Z=\{t:\ p(t)=0\}$. More precisely,

$\displaystyle |p(t)|\geqslant 2^{-O(d)}\left(\frac rR\right)^d,\qquad r=\mathrm{dist }(t,Z),\quad R=|t|>1.$

We expect that for an equation having only quasiunipotent Fuchsian singular points, the number of isolated roots of solutions can be explicitly bounded in terms of $n=\mathrm{ord }L,\ d=\max\deg p_k$ and $B=\angle L$. Indeed, it looks like we can cut out circular neighborhoods of all singularities and apply Theorem 1.

The trouble occurs when singularities are allowed to collide or almost collide. Then any slit separating them will necessarily pass through the area where the leading coefficient $p_0$ is dangerously small.

## 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.

# Local theory of regular singular points of linear systems

This lecture, in an exceptional way, will take place on Sunday, 16:00-18:00, in the Room 261.

1. Regular and irregular singularities: growth matters.
2. Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
3. Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
4. Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
5. Coincidence of formal and holomorphic classification in the Fuchsian case.
6. Integrability of the normal form.
7. Towards global theory of Fuchsian systems on $\mathbb C P^1$: Monopoles as special classes of rational matrix functions.

Reminder: Today (actually, on Friday) was the deadline for submission of the home exam 😦

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