Special cases of the embedding problem

Question

Embedding problem. Let $I$ be the ideal of polynomial algebra $A=K^{[n]}$, such that $A/I$ is also a polynomial algebra with smaller number $k$ of variables. Is it true that $I$ is generated by $n-k$ variables of $A$.

Definition. Call an ideal of polynomial algebra coordinate-like if it us generated by some coordinates of a polynomial algebra.

I am interested in the following particular cases of the embedding problem.

Problem 1. Consider any polynomial algebra $A=K^{[n]}$ and its $k$ polynomials $f_1,..., f_k$. For a polynomial algebra $B=K[y_1,..., y_{n+k}]$ consider the morphism sending $ y_i$ to $n$ coordinates of $A$ for $i\leq n$ and $y_j$ to $f_{j-n}$ for $j>n$. It is obvious that thr kernel of this morphism is the ideal $I$ with $B/I\cong A$. So is it coordinate-like?

Problem 2. If $I\subset J$ are two coordinate-like ideals of polynomial algebra $A$, is it true that $J/I$ is coordinate-like ideal of $A/I$?

It is obvious that these problems follow from the embedding problem, but are they true?

Answer 1

First, the embedding problem has a partial affirmative answer if $k\geq n-2$ or if $n\geq 2k+2$. In these cases, $I$ is generated by $n-k$ elements, not necessarily variables.

Problem 1 has an affirmative answer. $I$ is generated by $y_j-f_{j-n}(y_1,\ldots, y_n)$ for $j>n$ and these are coordinates of $B$.