Images of polynomial maps on large fields

Abstract

A field $k$ is called large if every irreducible $k$-curve with a $k$-rational smooth point has infinitely many $k$-points. Let $k$ be a perfect large field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. We will show that $k \setminus f_k(k)$ is infinite. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect large fields.