On Waring's problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m, n, k) such that (m + 1) generic skew-symmetric (k + 1)-forms in (n + 1) variables as linear combinations of the same s(m, n, k) decomposable skew-symmetric (k + 1)-forms. This problem is analogous to a well known problem called Waring's problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry. In this paper, we will go through some basics of algebraic geometry, describe how objects in algebraic geometry can be associated to systems of skew-symmetric forms, and discuss algebro-geometric approaches to establish the existence of triples (m, n, k), where s(m,n, k) is more than expected. (C) 2013 Elsevier Inc. All rights reserved.