On Waring's problem for many forms and Grassmann defective varieties (Q1855450)

Given positive integers \(n,d,k\) and \(h\), may we write any \(k+1\) homogeneous polynomials in \(n+1\) variables \(f_j(x_0,\dots,x_n)\), \(j=0,\dots,k\), of degree \(d\) as linear combinations of the same \(h+1\) powers of linear forms \(L_i(x_0, \dots,x_n)\), \(i=0,\dots,h\)? This is the simultaneous Waring problem for \(k+1\) forms. In characteristic zero a lemma of Terracini shows that this problem is equivalent to the equality \(\text{Sec}_{k,h} (V_{n,d})= G(n,r)\), where \(r:={n+d \choose n}-1\), \(G(k,r)\) is the Grassmannian variety of \(k\)-dimensional subspaces \(\mathbb{P}^r\), \(V_{n,d}\subset \mathbb{P}^r\) is the Veronese embedding of \(\mathbb{P}^n\) given by all homogeneous forms of degree \(d\) in \(n+1\) variables and, for any variety \(X\subset \mathbb{P}^r\), \(\text{Sec}_{k,h}(X) \subseteq G(k,r)\) denotes the closure of all \(k\)-planes contained in an \(h\)-plane spanned by \(h+1\) points of \(X\). The expected dimension of \(\text{Sec}_{k,h}(V_{n,d})\) is \(\min\{\dim (G(k,r)),(n+1)-1\}\) and hence a necessary condition is the inequality \((h+1)(n+k+1) \geq(k+1){n+d \choose n}\). Here the author uses the geometric translation to show that (in characteristic zero) this condition is sufficient if \(k\geq n\).