Embedding drawback. Let $I$ be the best of polynomial algebra $A=Okay^{[n]}$, such that $A/I$ can be a polynomial algebra with smaller quantity $okay$ of variables. Is it true that $I$ is generated by $n-k$ variables of $A$.

Definition. Name a super of polynomial algebra coordinate-like if it us generated by some coordinates of a polynomial algebra.

I’m within the following explicit circumstances of the embedding drawback.

Downside 1. Think about any polynomial algebra $A=Okay^{[n]}$ and its $okay$ polynomials $f_1,…, f_k$. For a polynomial algebra $B=Okay[y_1,…, y_{n+k}]$ contemplate the morphism sending $ y_i$ to $n$ coordinates of $A$ for $ileq n$ and $y_j$ to $f_{j-n}$ for $j>n$. It’s apparent that thr kernel of this morphism is the best $I$ with $B/Icong A$. So is it coordinate-like?

Downside 2. If $Isubset J$ are two coordinate-like beliefs of polynomial algebra $A$, is it true that $J/I$ is coordinate-like preferrred of $A/I$?

It’s apparent that these issues observe from the embedding drawback, however are they true?