Interpolation in affine and projective space over a finite field

Abstract

Let $s(n,q)$ be the smallest number $s$ such that any $n$-fold $\mathbb{ F}_q$-valued interpolation problem in $\mathbb{P}^k_{\mathbb{F}_q}$ has a solution of degree~$s$, that is: for any pairwise different $\mathbb{F}_q$-rational points $P_1,\ldots ,P_n$, there exists a hypersurface $H$ of degree~$s$ defined over $\mathbb{F}_q$ such that $P_1,\ldots ,P_{n-1}\in H$ and $P_n\notin H$. This function $s(n,q)$ was studied by Kunz and the second author in \cite{KuW} and completely determined for $q=2$ and $q=3$. For $q\geq 4$, we improve the results from \cite{KuW}.

The affine analogue to $s(n,q)$ is the smallest number $s=s_a(n,q)$ such that any $n$-fold $\mathbb{F}_q$-valued interpolation problem in $\mathbb{A}^k(\mathbb{F}_q)$, $k\in \mathbb{N}_{>0}$ has a polynomial solution of degree $\leq s$. We exactly determine this number.