On partial polynomial interpolation
This provides a complete solution to a fundamental problem in polynomial interpolation for algebraic geometers, extending a classical result to a more general setting.
The paper generalizes the Alexander-Hirschowitz theorem to arbitrary zero-dimensional schemes contained in a general union of double points, showing that the affine space of polynomials of degree ≤ d in n variables with assigned values of general linear combinations of first partial derivatives has the expected dimension for d ≠ 2 with only five exceptions, and fully describes the exceptional cases for d = 2.
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.