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This might be a silly question but is there a criterion for when the quotient of $k[X_1,\ldots, X_n]$ by some ideal is isomorphic to a polynomial ring? For instance $k[X,Y]/\langle Y \rangle \simeq k [X]$ but $k[X,Y]/\langle Y^2 \rangle$ is not a polynomial ring.

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An obvious necessary condition is that the ideal has to be prime, which is enough to rule out $\langle Y^2\rangle$. –  Emil Jeřábek Feb 8 '13 at 11:56
And the quotient ring has to be regular; check out mathoverflow.net/questions/79956/… –  Allen Knutson Feb 8 '13 at 12:16
I doubt that there are is a (useful and nontrivial) criterion. One could use it to attack the wide open Zariski cancellation problem. –  Martin Brandenburg Feb 8 '13 at 16:41
There are probably no such non-trivial criterion. One deep result is the Abhyankar-Moh theorem which says that $\mathbb{C}[x,y]/f$ is the polynomial ring in one variable if and only if there is a $\mathbb{C}$- algebra automorphism of $\mathbb{C}[x,y]$ which transforms $f$ into a variable. –  Mohan Feb 8 '13 at 17:07
Martin: I'm not sure the cancellation problem is so wide open any more. Check out arxiv.org/abs/1208.0483. –  Neil Epstein Feb 9 '13 at 8:35
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