A new proof of the Alexander-Hirschowitz interpolation Theorem

TL;DR

This paper presents a new proof of the Alexander-Hirschowitz interpolation theorem, which characterizes when collections of double points impose independent conditions on hypersurfaces, using degeneration techniques of projective space.

Contribution

The paper introduces a novel proof method for the Alexander-Hirschowitz theorem based on degenerations of projective space and analysis of linear system degenerations.

Findings

Validates the Alexander-Hirschowitz theorem with a new degeneration-based proof

Identifies the conditions under which double points impose independent conditions

Provides a framework for analyzing polynomial interpolation problems in multiple variables

Abstract

The classical polynomial interpolation problem in several variables can be generalized to the case of points with greater multiplicities. What is known, as yet, is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in P^r gives independent conditions on the linear system L of the hypersurfaces of degree d, with a well known list of exceptions. We present a new proof of this theorem which consists in performing degenerations of P^r and analyzing how L degenerates.