## Local theory of Fuchsian systems (cont.)

- Resonant normal form.

**Definition.**A meromorphic Fuchsian singularity , , is in the (Poincare-Dulac) normal form, if for all , the identities hold. **Theorem.**Any Fuchsian system is holomorphically gauge equivalent to a system in the normal form.- Integrability of the normal form: let (in fact, the sum is finite). Then the solution is given by the (non-commutative) product . The monodromy is the (commutative) product, .

References: [IY], section 16.

## Linear high order homogeneous differential equations

- Differential operators as noncommutative polynomials in the variable with coefficients in a differential field of meromorphic germs at the origin.
- Composition and factorization.
- Reduction of a linear equation to a system of linear first order equations and back. Singular and nonsingular equations.
- Euler derivation and Fuchsian equations (“nonsingular with respect to “).
- 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.

[…] 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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