Analytic Birkhoff reduction for Poincaré rank 1 systems with diagonalizable leading term

Determine whether every linear differential system dX/dz = A(z) X with Poincaré rank r = 1 and diagonalizable leading coefficient A0 (allowing eigenvalue multiplicities) is analytically equivalent—via a gauge transformation T(z) meromorphic at z = ∞ with T(∞) = I—to a Birkhoff standard form B(z) = z^{r-1} ∑_{p=0}^{r} B_p z^{-p}; specify necessary and sufficient conditions or construct an explicit procedure for such a reduction.

Background

The paper reviews Birkhoff’s reduction problem for linear ODE systems with Poincaré rank r, asking whether a system can be reduced to a standard form via meromorphic or analytic equivalence. For r = 1, they recall positive results: Turrittin proved meromorphic equivalence under distinct-eigenvalue assumptions on A0, and Birkhoff and Bolibrukh provided conditions for analytic equivalence in certain settings.

The authors then focus on rank r = 1 with A0 diagonalizable, present partial results when eigenvalues are distinct (showing meromorphic equivalence to the Birkhoff standard form), and discuss challenges when eigenvalues have multiplicities. They explicitly state that the case with diagonalizable leading term still remains open, and only outline a possible approach.