Linear system of hypersurfaces passing through a Galois orbit

Let d and n be positive integers, and E/F be a separable field extension of degree \(m=\left( {\begin{array}{c}n+d\\ n\end{array}}\right) \) $m=\left(\begin{array}{c}n+d\\ n\end{array}\right)$ . We show that if \(|F| > 2\) $|F|>2$ , then there exists a point \(P\in \mathbb {P}^n(E)\) $P\in {\mathbb{P}}^n(E)$ which does not lie on any degree d hypersurface defined over F. In other words, the m Galois conjugates of P impose independent conditions on the m-dimensional F-vector space of degree d forms in \(x_0, x_1, \ldots , x_n\) $x_0,x_1,\dots ,x_n$ . As an application, we determine the maximal dimensions of linear systems \(\mathcal {L}_1\) ${\mathcal{L}}_1$ and \(\mathcal {L}_2\) ${\mathcal{L}}_2$ of hypersurfaces in \(\mathbb P^n\) ${\mathbb{P}}^n$ over a finite field F, where every F-member of \(\mathcal {L}_1\) ${\mathcal{L}}_1$ is reducible and every F-member of \(\mathcal {L}_2\) ${\mathcal{L}}_2$ is irreducible.