Polynomial maps on vector spaces over a finite field

Abstract

Let $l$ be a finite field of cardinality $q$ and let $n$ be in ${\mathbb{Z}}_{\ge 1}$. Let $f_1,\dots ,f_n\in l[x_1,\dots ,x_n]$ not all constant and consider the evaluation map $f=(f_1,\dots ,f_n):l^n\to l^n$. 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.

6 pages