Sergei Yakovenko's blog: on Math and Teaching

Monday, November 10, 2014

Lecture 3 (Nov. 10, 2014).

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 […]

    Pingback by Lecture 12+ (Mon, Dec 22, 2014) | Sergei Yakovenko's blog: on Math and Teaching — Tuesday, December 30, 2014 @ 8:31 | Reply


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