On partial polynomial interpolation
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2007
- DOI:
- 10.48550/arXiv.0705.4448
- arXiv:
- arXiv:0705.4448
- Bibcode:
- 2007arXiv0705.4448C
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Numerical Analysis;
- 14C20;
- 15A72;
- 65D05;
- 32E30
- E-Print:
- 34 pages, 2 tables, revised version: different proof of Theorem 4.1, Section 4 significantly changed, Appendix added