On the Alexander-Hirschowitz Theorem

Abstract

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\mathbf{P}}^n$ 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.

29 pages, the proof in the case of cubics has been simplified, three references added, to appear in J. Pure Appl. Algebra