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 $m=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$. 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 $x_0,x_1,\dots ,x_n$. 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.

16 pages; Macaulay2 code included as a supplementary file to justify examples on pages 12-13