= Poincaré rank
{c}
{title2=$r=p-1$}
For a first-order meromorphic <linear ordinary differential equation> $Y_t=A(t)Y$, with a coefficient <pole> of order $p>1$ in a fixed local coordinate and gauge, its Poincaré rank is $p-1$. At infinity one must transform the differential system, not just the <matrix>-valued function: $Y_\lambda=A_0\lambda^mY+\cdots$ becomes $Y_t=-A_0t^{-m-2}Y+\cdots$ under $t=1/\lambda$, giving rank $m+1$. A singular gauge can lower an apparent rank; distinct leading <eigenvalues> prevent the elementary degeneracy in the unramified setting.
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