Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is not surjective, that is, $M_f(F) > 0$. Is the number $M_f(F)$ big?

If $F = {\mathbb F}_q$ is a finite field of $q$ elements and $f\in \mathbb{F}_q[x]$ is a polynomial of degree $d>0$, then a simple result shows that if $M_f(F_q) > 0$, then $$M_f(F_q) \geq \frac{q-1}{d}. $$ As $q$ goes to infinity, this lower bound goes to infinity. This suggests the following question for which I do not know the answer.

Question. Assume that $F$ is an infinite field. If $M_f(F)>0$. Is it true that $M_f(F) = \infty$? In other words, for an infinite field $F$, if $f \in F[x]$ misses one value, is it true that $f$ misses infinitely many values?

The answer is positive if $F$ is big in the sense that F is algebraically closed (${\mathbb C}$ etc), or topologically closed ($\mathbb{R}$ or $\mathbb{Q}_p$ etc) or if $F$ is a small field such as a Hilbertian field (number fields etc). Anyone has more examples or counter-examples?

Gjergji-Zaimi points out that the above question has been asked before, see Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

To make my question somewhat newer, I will ask a multi-variable version of the question. It is unknown to me even in the case that the field is the complex numbers.

Question: Assume that $f=(f_1,..., f_n): F^n -> F^n$ is a FINITE polynomial map in n variables over an infinite field F. If f misses one value, is it true that f misses infinitely many values?

I do not know the answer even in the case when $F={\mathbb C}$ is the complex numbers if $n \geq 2$. Note that the finiteness assumption of the map $f$ cannot be dropped, as Kharlamov (2012) has given the example $$f=(u(uv-1), u^2-(uv-1)^2)$$ which misses precisely one point, the origin (0,0) of ${\mathbb C}^2$.