Polynomial maps on vector spaces over a finite field

TL;DR

This paper establishes a new lower bound on the size of the complement of the image of polynomial maps over finite fields, improving previous bounds and contributing to the understanding of polynomial mappings in finite field vector spaces.

Contribution

The paper provides an improved lower bound on the number of elements not attained by polynomial maps over finite fields, refining earlier results in the field.

Findings

Derived a new lower bound for the size of the complement of polynomial map images.

The bound depends on the degree of the polynomial map and the size of the finite field.

The result generalizes and improves upon previous bounds in the literature.

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.