On partial polynomial interpolation
Abstract
The AlexanderHirschowitz 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 zerodimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2007
 DOI:
 10.48550/arXiv.0705.4448
 arXiv:
 arXiv:0705.4448
 Bibcode:
 2007arXiv0705.4448C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Numerical Analysis;
 14C20;
 15A72;
 65D05;
 32E30
 EPrint:
 34 pages, 2 tables, revised version: different proof of Theorem 4.1, Section 4 significantly changed, Appendix added