  In this paper we study the singularity of multivariate Hermite interpolation
of type total degree. We present a method to judge the singularity of the
interpolation scheme considered and by the method to be developed, we show that
all Hermite interpolation of type total degree on $m=d+k$ points in $\R^d$ is
singular if $d\geq 2k$. And then we solve the Hermite interpolation problem on
$m\leq d+3$ nodes completely. Precisely, all Hermite interpolations of type
total degree on $m\leq d+1$ points with $d\geq 2$ are singular; for $m=d+2$ and
$m=d+3$, only three cases and one case can produce regular Hermite
interpolation schemes, respectively. Besides, we also present a method to
compute the interpolation space for Hermite interpolation of type total degree.
