Hypersurfaces passing through the Galois orbit of a point

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $\mathrm{\mid }K\mathrm{\mid }>2$, then for any positive integers $d$ and $n$, and separable field extension $L\mathrm{/}K$ with degree $m=\left(\genfrac{}{}{0px}{}{n+d}{d}\right)$, there exists a point $P\in {\mathbb{P}}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $\mathrm{\mid }K\mathrm{\mid }=2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L\mathrm{/}K$ with degree $r$, there exists a point $P\in {\mathbb{P}}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.