This is a follow-up of my previous question, but it is self-contained.
Let $f\in k[x_i,i=0,\ldots ,n]$ be a a polynomial which contains $x_0$ (i.e. polynomial like $f=x_1$ is not allowed), where $k$ is a field with characteristic $0$. We define a map $$Q_a: k[x_i,i=0,\ldots ,n] \to k[x_i,i=1,\ldots ,n]$$ which sends $f(x_0,\ldots,x_n)$ to $f(a_1x_1+\ldots+a_n x_n, x_1,\ldots,x_n)$, where $a=(a_1,\ldots a_n)$ is a parameter.
Now my question is
Is it true that there exists a neighborhood $U_a$ of $a$, such that for all $U_a\ni a '\neq a$,we have $Q_a(f)\neq Q_{a'}(f)$?
I think one way is to show for fixed $a$ there is only finitely many $a'$ with $Q_a(f)= Q_{a'}(f)$.