Extending linear and quadratic functions from high rank varieties

Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety. We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional linear subspace $W$ of $V$ contained in $X$ is linear and that it is weakly quadratic if its restriction to any three-dimensional linear subspace $W$ of $V$ contained in $X$ is quadratic. We say that $X$ is admissible if any weakly linear function on $X$ is a restriction of a linear function on $V$ and any weakly quadratic function on $X$ is a restriction of a quadratic function on $V$. The main result in the paper concerns the case when the field $k$ is a finite. We show that for any $d,L\geq 1$ there exists $r=r(d,L,k)\in \mathbb Z _+$ such that any complete intersection $X\in V$ in a vector space $V$ of codimension $L$, degree $d$ and rank $\geq r$ is admissible. Moreover we show the existence of a function $r(d,L)$ such that one can take $r(d,L,k)=r(d,L)$ for all finite fields $k$ of characteristic $>d$. The proof of the admissibility for finite fields $k$ is based on bounds on the number of $k$-points on ancillary varieties $E(X)$. These results allow us to bound the dimension of varieties $E(X)$. Using these results we were able to prove the admissibility of complex homogeneous varieties of high rank. Using the results of \cite{br} one can extend our proofs to show the admissibility of varieties of high rank over local non-archimedian fields. Also using Corollary $4.3$ of \cite{cmpv} one can dispense with the assumption that $X$ is a complete intersection.