Extending weakly polynomial functions from high rank varieties

Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this paper we consider the case when $X= \mathbb X (k)$ where $\mathbb X$ is a complete intersection of bounded codimension defined by a high rank polynomials of degrees $d, char(k)=0$ or $char (k)>d$ and either $k$ is algebraically closed, or $k=\mathbb F _q,q>ad$. We show that under these assumptions any $k$-valued weakly polynomial function of degree $ \leq a$ on $X$ is a restriction of a polynomial of degree $\leq a$ on $V$. Our proof is based on Theorem 1.11 on fibers of polynomial morphisms $P:\mathbb F _q^n\to \mathbb F _q^m$ of high rank. This result is of an independent interest. For example it immediately implies a strengthening of the result of [4].