In this article we study the following problem. Let $k$ be an infinite field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. Is $k \smallsetminus f_k(k)$ infinite? We give a positive answer to this question when $k$ is a perfect ample field. In fact, we prove that $|k \smallsetminus f_k(k)|=|k|$. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect ample fields.
"Images of polynomial maps on ample fields." Funct. Approx. Comment. Math. 55 (1) 23 - 30, September 2016. https://doi.org/10.7169/facm/2016.55.1.2