Let $a_1,\ldots,a_m \in \mathbb{C}[x_1,\ldots,x_n]:=S$, with $m > n \geq 2$.
Denote $R:=\mathbb{C}[a_1,\ldots,a_m]$ and assume that $I$ is an ideal of $R$ such that $R/I \cong \mathbb{C}[x_{r+1},\ldots,x_n]$, for some $1 \leq r \leq n-1$.
If I am not wrong, there exists an automorphism $f$ of $S$ such that $f(I) \subseteq Sx_1+\cdots+Sx_r$ (= the ideal in $S$ generated by $x_1,\ldots,x_r$).
Question: Is it true that there exists an automorphism $g$ of $S$ such that the images of the generators $a_1,\ldots,a_m$ under $g$, $g(a_1),\ldots,g(a_m)$, belong to $\mathbb{C}[x_1,\ldots,x_r]$?
Example: $a_1=x^2, a_2=x^3, a_3=y$, $R=\mathbb{C}[x^2,x^3,y]$, $S=\mathbb{C}[x,y]$. Take $I=Rx^2+Rx^3$. $R/I$ is isomorphic to $\mathbb{C}[y]$. ($n=2$, $m=3$, $r=1$). Here $I \subseteq Sx$ and the generators of $I$, $x^2$ and $x^3$, belong to $\mathbb{C}[x]$. Here $f=g$ the identity map: $f(s)=s$ for every $s \in S$.
Thank you very much!