# Sergei Yakovenko's blog: on Math and Teaching

## 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).

## Finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$, I. Formal theory.

1. Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$. Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$.
4. Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$: every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

Disclaimer applies, as usual 😦

## Wednesday, November 21, 2007

### “Auxiliary Lesson” שעור עזר) #5) November 22, 2007

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 5:41
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## Formal linearization and obstructions. Poincare theorem

1. Formal equivalence of formal vector fields (total recall)
3. Poincare formal linearization theorem
4. Proof of the Poincare theorem:
• Homological equation
• Commutator with diagonal linear vector field
• Stabilization of the series
5. Resonant monomials. Resonant normal form. Poincare–Dulac paradigm.
6. Formal classification of formal self-maps. Multiplicative resonances.
7. Survey of further results. Formal types of line and planar singularities.

Reading material: Section 4 from the book (printing disabled).Disclaimer (alas, still required) .

## Wednesday, November 14, 2007

### “Auxiliary Lesson” שעור עזר) #4) November 15, 2007

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 8:43
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## Formal series, formal vector fields: Flows and embedding

1. Formal series algebra $\mathbb C[[x_1,\dots,x_n]]$. Formal vector fields (derivations) $\mathscr D[[\mathbb C^n,0]]$. Formal equivalence of vector fields. Truncation. Convergence in the formal algebra  $\mathbb C[[x_1,\dots,x_n]]$.
2. Formal inverse function theorem. Geometric series.
3. Integration and formal flow of vector fields. Exponent.
4. Embedding in the flow. Linear case. Matrix logarithms.
5. Embedding in the flow and formal logarithms.

Reading Section 3 from the textbook.

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