Sergei Yakovenko's blog: on Math and Teaching

Tuesday, November 18, 2014

Lecture 5 (Nov. 17)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 12:27
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Fuchsian equivalence and Fuchsian classification

Definitions
A (formal, genuine) Fuchsian operator of order n is a (formal, resp., converging) series of the form L=\sum_{k=0}^\infty t_kp_k(\epsilon) with the coefficients p_k\in\mathbb C[\epsilon] from the ring of polynomials in the variable \epsilon with n=\deg p_0\ge \deg p_k\ \forall k=1,2,\dots.
The polynomial p_0\in\mathbb C[\epsilon] is called the Euler part of L.

Two Fuchsian operators L,M are \mathscr F-equivalent (formally or analytically), if there exist two Fuchsian operators H,K such that MH=KL and the Euler parts of H,L are mutually prime in \mathbb C[\epsilon].


Unlike the Weyl algebra \mathscr M(\mathbb C,0)[\epsilon], the collection of Fuchsian operators is not a subalgebra, although it is “multiplicatively” (compositionally) closed.

The Fuchsian equivalence is indeed reflexive, transitive and symmetric. The first two properties are obvious, to prove the last one an additional effort is required. Indeed, for two Fuchsian operators L,H of order n with mutually prime Euler parts, one can construct two operators U,V with holomorphic coefficients so that the identity UL+VH=1 holds, but in the leading terms of U,V may well degenerate, thus violating the Fuchsian condition. However, one can always find such pair of operators of order greater by 1, which will still be Fuchsian. The rest is easy.


The following results can be proved by more or less direct computation in the algebra \mathscr W=\mathbb C[[t]]\otimes\mathbb C[\epsilon]:

  • A Fuchsian operator with a nonresonant Euler part is \mathscr F-equivalent to its Euler part.
  • Any Fuchsian operator is \mathscr F-equivalent to a polynomial operator from \mathbb C[t]\otimes\mathbb C[\epsilon].
  • Any Fuchsian operator is \mathscr F-equivalent to a polynomial operator of the form L=(\epsilon -\lambda_1+q_1(t))\cdots(\epsilon -\lambda_n+q_n(t)) with q_1(t),\dots,q_n(t) being polynomials without free terms, q_i(0)=0, which is Liouville integrable.
  • A Fuchsian operator has trivial (identical) monodromy if and only if it is \mathscr F-equivalent to an Euler operator with pairwise different integer roots. The corresponding equation has an apparent singularity (all solutions are analytic) if and only if all these roots are pairwise different nonnegative integers.

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