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I am pondering over the following question: Suppose $k$ is a field and we are given a set $S\subseteq k[x_i,...,x_n]$ of polynomials. When is the ring $k[S]$ where we adjoin all the elements of $S$ to $k$ a polynomial ring on the set $S$?

I have made the following observations:

  • of course one should have that $S$ contains $n$ elements or less, otherwise we can find some relations.

  • something like $S=\{x^2,x^3\}\subset k[x]$ does not work because $(x^2)^3=(x^3)^2$.

  • something like $S=\{x,y,xy\}\subset k[x,y,z]$ does not work because xy=x*y.

So my guess is, that the answer is that $S=\{s_1,...\}$ has to be such that $s_i$ is not already an element in $k[s_1,...,s_{i-1}]$. Are there ''easier" criterions for this? For some ideal $J\subset k[x_1,...,x_n]$, can one always extract a generating set $S=\{a_1,...,a_c\}$ such that $k[a_1,...,a_c]$ is a polynomial ring? Under which assumptions is this possible, if this is not possible in general?

slin0
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  • I'm not clear what you are asking. There will always be a smallest subring of $k[x_1,\ldots,x_n]$ that contains both field $k$ and subset $S$. What leads you to the claim "that $S$ contains $n$ elements or less, otherwise we can find some relations"? What would be wrong with finding relations in a polynomial ring? – hardmath Feb 23 '19 at 18:24
  • Your guess is wrong: $s_i$ might be algebraic over $k[s_1,\dots,s_{i-1}]$ without being an element of it. – Eric Wofsey Feb 23 '19 at 18:43
  • @hardmath If I choose $S={x^2}$ I would obtain $k[x^2]$ which I can consider as a polynomial ring with variable $x^2$. If I choose something like $S={x^2,x^3}$, then I cannot view $k[x^2,x^3]$ as a polynomial ring on $x^2,x^3$ as we have the relation mentioned above. Of course I could view $x^2$ and $x^3$ as formal variables but this is not what I want. – slin0 Feb 23 '19 at 18:45
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    Near-duplicate: https://math.stackexchange.com/questions/730274/how-to-determine-a-set-of-polynomials-is-algebraically-indepedent-or-not – Eric Wofsey Feb 23 '19 at 18:50
  • Also https://mathoverflow.net/questions/41535/how-to-show-a-set-of-polynomials-is-algebraically-independent – Eric Wofsey Feb 23 '19 at 18:51

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