  The Alexander-Hirschowitz theorem says that a general collection of $k$
double points in ${\bf P}^n$ 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 $d\neq
2$ with only five exceptional cases. If $d=2$ the exceptional cases are fully
described.
