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.