On partial polynomial interpolation

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\mathbf{P}}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree $\le d$ in $n$ variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if $d\mathrm{\ne }2$ with only five exceptional cases. If $d=2$ the exceptional cases are fully described.