Linear system of hypersurfaces passing through a Galois orbit

Abstract

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ 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 $x_0, x_1, \ldots, x_n$. As an application, we determine the maximal dimensions of linear systems $\mathcal{L}_1$ and $\mathcal{L}_2$ of hypersurfaces in $\mathbb P^n$ over a finite field $F$, where every $F$-member of $\mathcal{L}_1$ is reducible and every $F$-member of $\mathcal{L}_2$ is irreducible.