UBC Number Theory Seminar: Shamil Asgarli

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 $P^{n}$. The dimension of this parameter space is $m=(n+dd)$. If $P_1, ..., P_m$ are in "general position", then no hypersurface of 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 $P_1, ..., P_m$ so that:

(a) $P_1, P_2 ..., P_m$ 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 $P_1, P_2, ..., P_m$.

We show that the answer is "Yes" if the base field $K$ has at least 3 elements. In other words, the concept of "general position" for points can be modelled by Galois orbits. As an application, we compute the maximum dimension of a linear system of hypersurfaces over a finite field $F_q$ where each $F_q$-member of the system is irreducible over $F_q$. This is joint work with Dragos Ghioca and Zinovy Reichstein.