Consider a polynomial algebra $A=\mathbb{K}[x\_1,\ldots,x\_n]$ and its ideal $I$, such that $A/I=\mathbb{K}[y\_1,\ldots, y\_k]$. Is it true that there exist new polynomial generators $z\_1,\ldots,z\_n$ (in a sense that $A=\mathbb{K}[z\_1,\ldots,z\_n]$) such that $I=(z\_1,\ldots,z\_{n-k})$?



Polynomial algebra and its special ideals
This is called the Embedding Problem. I believe that it is false in positive characteristics and unknown in characteristic zero, except a few small cases. See [Kraft's review](https://mathscinet.ams.org/mathscinet-getitem?mr=1423629)
for the extent of my knowledge of the state of the problem. The counterexample in characteristic $p$ is from this paper:
$$
K[x,y]/(y^{p^2}-x-x^{2p}) 
$$


