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.