On the Alexander-Hirschowitz Theorem

TL;DR

This paper presents a self-contained proof of the Alexander-Hirschowitz theorem, which characterizes when collections of double points impose independent conditions on polynomials, with a focus on simplifying the proof for degree 3 cases.

Contribution

The paper provides a simplified, self-contained proof of the Alexander-Hirschowitz theorem, especially streamlining the case for degree 3, and discusses the historical development of the problem.

Findings

Confirmed the list of exceptions for the theorem

Provided a shorter proof for the case d=3

Clarified the historical context of the theorem

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. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case $d=3$, where our proof is shorter. We end with an account of the history of the work on this problem.