On partial polynomial interpolation

TL;DR

This paper generalizes the Alexander-Hirschowitz theorem to include arbitrary zero-dimensional schemes in general unions of double points, providing a comprehensive understanding of polynomial interpolation with derivatives.

Contribution

It extends the classical theorem to broader schemes and characterizes the dimension of polynomial spaces with derivative conditions, including detailed exceptions.

Findings

The affine space of polynomials has the expected dimension in most cases.

Five exceptional cases occur when the degree is not 2.

All exceptions for degree 2 are fully described.

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.