  For a class of ordinary differential operators $P$ with polynomial
coefficients, we give a necessary and sufficient condition for $P$ to be
globally regular in $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply
$u\in \cS(\R)$ (this can be regarded as a global version of the Schwartz'
hypoellipticity notion). The condition involves the asymptotic behaviour, at
infinity, of the roots $\xi=\xi_j(x)$ of the equation $p(x,\xi)=0$, where
$p(x,\xi)$ is the (Weyl) symbol of $P$.
