On Waringʼs problem for systems of skew-symmetric forms

Abstract

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.