Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic polynomial: $f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$, $a_j \in \mathbb{C}[x,y]$. Denote: $A=\mathbb{C}[x,y]$ and $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][w]$, where $w^n+a_{n-1}w^{n-1}+\cdots+a_1w+a_0=0$.
If I am not wrong, the following claim is mentioned in the comments to this question:
Claim: If $n \geq 4$, then $B$ is a UFD for 'sufficiently general $f$'.
Question 1: Could one please describle those 'sufficiently general $f$'s'?
Question 2: What if $f$ is not monic?
Thank you very much!