Consider the vector space (parameter space) of all homogeneous forms of degree d in n+1 variables defined over some field K. Geometrically, the vanishing set of such a form corresponds to a hypersurface of degree d in the projective space Pn. The dimension of this parameter space is m = 􏰁n+d􏰂. If P1, ..., Pm are in ”general position”, then no hypersurface of d degree d can pass through all these m points, because passing through each additional point imposes 1 new linearly independent condition. In this talk, we address the following variant: for a given K, d, and n, can we always find m points P1,...,Pm so that: (a) P1,P2...,Pm form a Gal(L/K)-orbit of a single point P defined over a Galois extension L/K with [L : K] = m, and (b) No hypersurface of degree m defined over K passes through P1,P2,...,Pm. We show that the answer is ”Yes” if the base field K has at least three elements. In other words, the concept of ”general position” for points can be modeled by Galois orbits. As an application, we compute the maximum dimension of a linear system of hypersurfaces over a finite field Fq where each Fq-member of the system is irreducible over Fq. This is joint work with Dragos Ghioca and Zinovy Reichstein.