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What are the elements in $k[X,Y]/(X^2-Y^3)$ like, where $k$ is a field?

For example, in $k[X]/(x^2+2x+3)$, all elements are of a degree lower than $2$. But I can't quite figure out the multi-variable case.

My first guess was that we could treat $Y$ as a constant and ensure that all the elements of $k[X,Y]/(X^2-Y^3)$ had their degree of $X$ as less than $2$. $Y$ obviously could then have any degree.

But then again, we could do also ensure that all elements had their degree of $Y$ as less than $3$, letting $X$ take any degree.

Having two representations for the elements of $k[X,Y]/(X^2-Y^3)$ sounds a little spurious to me.

Any help would be much appreciated.

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    $\begingroup$ But you already have infinitely many representations of every element of $k[X]/(x^2+2x+3)$; you can take one and add on any multiple of $x^2+2x+3$. OK, in this case you can pick the one of smallest degree, but there's nothing particularly canonical about that. In the two variable case it's just harder to decide which is the best representation. $\endgroup$ – mdp Jan 31 '14 at 11:59
  • $\begingroup$ Are you saying my hunch regarding the elements of $k[X,Y]/(X^2-Y^3)$ is correct? $\endgroup$ – algebraically_speaking Jan 31 '14 at 12:00
  • $\begingroup$ Insofar as I understand what your hunch is, yes. It is possible to minimize either the degree of $X$ or that of $Y$, to within the bounds that you give, and also to do other things entirely. $\endgroup$ – mdp Jan 31 '14 at 12:05
  • $\begingroup$ An entirely different answer to the question in your title, which may also help, is the following. Let $X$ and $Y$ be coordinates on $k^2$, so that $k[X,Y]$ is identified with the set of polynomial functions on $k^2$. Then $k[X,Y]/(X^2-Y^2)$ is the restriction of this set of polynomial functions to the subset of $k^2$ satisfying $X^2-Y^3=0$. Now two functions differing by a multiple of $X^2-Y^3$ define the same function when restricted to this subset, so are identified. $\endgroup$ – mdp Jan 31 '14 at 12:08
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There is nothing spurious here, and both of your representations of the elements are perfectly fine. In general, choosing a basis for the elements of a $k$-algebra will depend on some arbitrary choices.

For example, one standard way of choosing a basis is the technique of Gröbner bases, which depends (in your example) on a choice of order for the monomials of $k[X, Y]$. There are different choices, which give different bases. And in fact, the two representations you give can be regarded as coming from two different Gröbner bases of your example.

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You can think of $k[x,y]/(x^2 - y^3)$ as the subring $k[t^2,t^3]$ inside $k[t]$. Let $k[x,y] \mapsto k[t]$ by $x \mapsto t^3$ and $y \mapsto t^2$, and more generally $f(x,y) \mapsto f(t^3,t^2)$. This substitution map is a $k$-algebra homomorphism and its image is clearly $k[t^2,t^3]$. Check that the kernel is precisely the multiples of $x^2 - y^3$ in $k[x,y]$, so we get the desired $k$-algebra homomorphism.

Note $k[t^2,t^3]$ is strictly smaller than $k[t]$ since it does not contain $t$. The ring $k[t^2,t^3]$ is $k + t^2k[t]$: all polynomials in $k[t]$ with linear coefficient equal to $0$.

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