# Sergei Yakovenko's blog: on Math and Teaching

## 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.

## 1 Comment »

1. […] local realization of the monodromy is by no means unique: in the non-resonant case one can realize any matrix by an Euler system with the eigenvalues which can be arbitrarily […]

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