Polynomial maps on vector spaces over a finite field

Abstract

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set $\mathrm{deg}(f)=\max_i \mathrm{deg}(f_i)$. Assume that $l^n \setminus f(l^n)$ is not empty. We will prove \begin{align*} |l^n\setminus f(l^n)| \geq \frac{n(q-1)}{\mathrm{deg}(f)}. \end{align*} This improves previous known bounds.