On Waringʼs 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 Waringʼs 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.