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 = n + d n . We show that if | F | > 2 , then there exists a point P ∈ 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 , ... , x n . As an application, we determine the maximal dimensions of linear systems L 1 and L 2 of hypersurfaces in P n over a finite field F, where every F-member of L 1 is reducible and every F-member of L 2 is irreducible. [ABSTRACT FROM AUTHOR]