On partial polynomial interpolation

Details On partial polynomial interpolation On partial polynomial interpolation

2007-05-30

arxiv.org/abs/0705.4448v3

Mathematics

Algebraic Geometry

34 pages, 2 tables, revised version: different proof of Theorem 4.1, Section 4 significantly changed, Appendix added

Scientific paper

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.

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