The rank of a linear differential
system in the neighborhood of a pole of the system is defined to be one less than the
order of the pole of the coefficient matrix of the system. H. L. Turrittin has shown
that arbitrary rank can be reduced to rank one at the expense of increasing the
dimension of the system in proportion to the amount of reduction. Can this
procedure lead to extraneous solutions of the rank-reduced system which differ in
behavior from solutions of the given system? This question is answered by a
transformation of the rank-reduced system to a block-diagonal form, exhibiting the
precise relation between solutions of the two systems. In particular, if the original
system has a regular singularity at the pole in question, then so does the
rank-reduced system.
