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I was wondering how to write $x$ as an algebraic combination of $\{x+xy,y+xy,x^2,y^2\}$, with the coefficients $\in \mathbb Q[x,y]$.

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I think that the kind of combination that you want is called an algebraic combination. – alex.jordan Feb 5 '12 at 0:58
I've revised my answer to include a solution to this new, edited problem. See my post below. – KReiser Feb 5 '12 at 0:59
@alex.jordan Thanks. – John Conn Feb 5 '12 at 1:00

New problem: Take $(x+xy)-x(y+xy)+y(x^2)$. This simplifies to $x+xy-xy-x^2y+x^2y=x$.

Old problem: This is not possible. To see why, examine $x= a(x+xy)+b(y+xy)+cx^2+dy^2$ with $a,b,c,d\in\mathbb{Q}$. We then have that $x= ax+by+(a+b)xy+cx^2+dy^2$. Equating coefficients, we see that $b$ must be 0 and $a$ must be 1. But then the coefficient of $xy$ is 1 on the RHS and 0 on the LHS, which is a contradiction.

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Cool! Thanks a lot. – John Conn Feb 5 '12 at 1:01

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