Veroneseans, power subspaces and independence (Q2844290)

Let \(d,n>1\) be integers and let \(N=\binom{d+n-1}{d}\). For any field \(K\), consider the (vector) Veronese map \(v_d:K^n\to K^N\). A \(1\)-dimensional subspace of \(v_d(K^n)\) is called a Veronesean point of degree \(d\).NEWLINENEWLINE The authors show that any \(d+1\) Veronesean points of degree \(d\) in \(K^N\) are independent. Moreover, they prove a generalization of this result for sets of subspaces of \(v_d(K^n)\). If the characteristic of \(K\) is suitable, then Veronesean points or subspaces are closely related to so-called powerpoints or power subspaces, respectively. Thus, independence results of these are also obtained.