question on integer polynomial sequences

Question

Let $f:{\mathbb Z}\to {\mathbb Z}$ be a degree $d$ polynomial, and for $c\in{\mathbb Z}$ set $S_c=|\{n\in{\mathbb Z}:f(n)=c\}|$. It is easy that $S_c\le 2$ for all but finitely many $c$ (but not more, as $f$ could be a translated even polynomial), also clearly $S_c\le d$ for all $c$. Can one say something more, like, $\sum_{S_c>2}S_c=O(d)$, at least can the number of $c$ with $S_c>2$ be bounded in terms of $d$? Has there been any related research? Thanks a lot.

Answer 1

A quick search finds a cubic, $y=x^3-8281x$, with nine different $c$.

$\begin{array}{c|ccc} 'c'&'x1'&'x2'&'x3'\\ -288120& -105& 49& 56\\ -263640 & -104& 39 & 65\\ -150480 & -99 & 19 & 80\\ -89760 & -96 & 11 & 85\\ 0& -91 & 0 & 91\\ 89760 & -85 & -11 & 96\\ 150480 & -80 & -19 & 99\\ 263640 & -65 & -39 & 104\\ 288120 & -56 & -49 & 105 \end{array}$