On partial polynomial interpolation

The Alexander–Hirschowitz theorem says that a general collection of k double points in Pn 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 ⩽d in n variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if d≠2 with only five exceptional cases. If d=2 the exceptional cases are fully described.