Fuchsian singular point

A singular point $t=t_*$ of a linear system of first order ordinary differential equations $$\dot z=A(t)z,\qquad z\in\CC^n,\quad t\in (\CC,t_*)$$ is called Fuchsian (or Fuchsian singularity), if the matrix of coefficients $A(t)$ is defined in a punctured neighborhood $(\CC,t_*)$ of the point $t_*$ has a first order pole at this point.

A Fuchsian singularity is always regular, but not vice versa.

By a suitable holomorphic gauge transformation $z=H(t)w$ with a holomorphic invertible matrix function $H(t)\in\operatorname{GL}(n,\CC)$, $t\in(\CC,t_*)$, a Fuchsian singular point can be always brought into a polynomial normal form. To describe this form, assume for simplicity that $t_*=0$ and denote by $\lambda_1,\dots,\lambda_n$ the (complex) eigenvalues of the residue matrix $A_0=\lim_{t\to 0}tA(t)\in\operatorname{Mat}(n,\CC)$. The collection of eigenvalues is called non-resonant if $\lambda_i-\lambda_j\notin\NN=\{1,2,\dots\}$.

In the non-resonant case the Fuchsian system is locally gauge equivalent to the Euler system $\dot w=\frac1t A_0\cdot w$, with the residue matrix $A_0$ which can be assumed in the Jordan normal form (upper triangular). In the resonant case the normal form is polynomial, $$ \dot w=\frac1t(A_0+tA_1+\dots+t^d A_d)w, \qquad A_0=\text{upper triangular},\quad A_1,\dots,A_d\in\operatorname{Mat}(n,\CC), $$ where the (constant) matrix coefficient $A_k$ may containg a nonzero term in the $(i,j)$th position only if $\lambda_i-\lambda_j=k\in\NN$.