On partial polynomial interpolation

The Alexander-Hirschowitz theorem says that a general collection of k double points in P 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 6= 2 with only five exceptional cases. If d = 2 the exceptional cases are fully described. AMS Subject Classification: 14C20; 15A72; 65D05; 32E30 Both authors are partially supported by Italian MUR and are members of GNSAGAINDAM.