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? 
 
It is pointed out that the above question has been asked before, see 
http://mathoverflow.net/questions/6820/can-a-non-surjective-polynomial-map-from-an-infinite-field-to-itself-miss-only-fi


To make this question new, 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 cannot 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 ${\mathbb C}^2$.