On the extension of a class of Hermite multivariate interpolation problems
Hakop Hakopian, Anush Khachatryan
TL;DR
This paper solves the Hermite multivariate interpolation solvability problem for total multiplicities up to $2n+2$, extending the previous $2n+1$ result and linking to Severi's classical theorem. It establishes a sharp, geometry-driven criterion: $(\mathcal{N},\mathcal{X})^k$ is $n$-solvable exactly when no line contains more than $n+1$ interpolation data points and no conic contains more than $2n+1$ data points, under a $(\le)$-scheme assumption. The method reduces to the plane case ($k=2$) and proceeds by induction on $n$, employing reductions along lines and conics via $\Delta$-incidence schemes and the hat construction to preserve exactness and solvability, with a projection argument to extend to higher dimensions. The results are sharp (counterexample for $2n+3$) and connect to classical Severi-type results and their multivariate extensions, enriching the theory of Hermite interpolation and its algebraic-geometric underpinnings.
Abstract
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).