Interpolation with \(N\) variables (Q1176664)

Given points \(a_ 1,\dots,a_ \ell\) and non-zero vectors \(v_ 1,\dots,v_ \ell\) of \(k^ n\) (\(k\) a field) and positive integers \(n_ 1,\dots,n_ \ell\), there is proved an interpolation theorem giving conditions for the existence of a polynomial in \(n\) variables of degree at most \(d\) with prescribed \(v_ i\)-directional derivatives at \(a_ i\) of orders \(0,\dots,n_ i-1\), for \(i=1,\dots,\ell\).