  Linear recurrence operators in characteristic $p$ are classified by their
$p$-curvature. For a recurrence operator $L$, denote by $\chi(L)$ the
characteristic polynomial of its $p$-curvature. We can obtain information about
the factorization of $L$ by factoring $\chi(L)$. The main theorem of this paper
gives an unexpected relation between $\chi(L)$ and the true singularities of
$L$. An application is to speed up a fast algorithm for computing $\chi(L)$ by
desingularizing $L$ first. Another contribution of this paper is faster
desingularization.
