Abstract :
The classical Waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of sdth powers of linear forms. We show that its solution is implied by a result of J. Alexander and A. Hirschowitz, concerning the Hilbert functions of the ideal of functions Vanishing to order two at a generic set of s points in imager − 1. Using Macaulay′s inverse systems, we show that the Alexander-Hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are dth powers of a given set of general linear forms. We also determine the dimension of the family of degree d forms that have additive decompositions of length s. We then study several notions of length for forms f, having to do with the kind of length-s, zero-dimensional schemes Z in imager − 1 whose defining ideal I(Z) annihilates the inverse system of f. When Z is to consist of distinct points, we obtain the above length of additive decomposition of f. When Z is smoothable we obtain the "smoothable length" of f; when Z is arbitrary, we obtain a "scheme length" of f. All these lengths are at least as large as the dimension of the vector space of all order-i partial derivates of f, for each i. The above-mentioned length functions are distinct. Using results about the existence of nonsmoothable Gorenstein point singularities in codimension 4, we show that when r = 5 there are forms f of scheme length s, which are not in the closure of the family of forms having additive decompositions of length s. Finally, we propose a new set of Waring problems for forms, using these lengths.
