On the extension of a class of Hermite multivariate interpolation problems

We characterize the sets of solvability for Hermite multivariate interpolation problems when the sum of multiplicities is at most $2n+2$, with $n$ the degree of the polynomial space. This result extends an earlier theorem (2000) by one of the authors concerning the case $2n+1$. The latter theorem, in turn, can be regarded as a natural extension of a classical Theorem of Severi (1921).