A new proof of the Alexander-Hirschowitz interpolation Theorem

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.