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How can I find the inverse of the following bijective function:

$$ F(x,y) = (x-y^2, x + y - y^2) $$ I usually do inverses by using the matrix method, but I have no clue on how to face this problem

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You will find the inverse of $F$ if you express $x$ and $y$ from the equations $$u = x-y^2\quad\text{and}\quad v = x+y-y^2\text{.}$$ I believe the shortest way is to subtract these equations and proceed.

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Just write the following system:



Now you have $\xi$ and $\eta$ written in terms of $x$ and $y$... you have to do the converse, i.e. write $x$ and $y$ in terms of $\xi$ and $\eta$.

Subtracting the first equation from the second you find


and substituting this in the first, leads you to $\xi=x-y^2=x-(\eta-\xi)^2$ from which you have


Hence the inverse of $F$ is $$ F^{-1}(\xi,\eta)=(\xi+\eta^2+\xi^2-2\xi\eta,\; \eta-\xi)\;\;. $$ In order to convince yourself that this is correct just check (with a simple direct computation) that $(F\circ F^{-1})(\xi,\eta)\equiv(\xi,\eta)$ and $(F^{-1}\circ F)(x,y)\equiv(x,y)$ (it's only a computation).

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