F(x,y)=(x^{2}-y^{2}, 2xy)

**Biscaim** · March 21st 2009, 12:23 PM

How do I show that the map $F(x,y)=(x^{2}-y^{2}, 2xy)$ is injective for y>0?

**running-gag** · March 21st 2009, 12:41 PM

Originally Posted by Biscaim

How do I show that the map $F(x,y)=(x^{2}-y^{2}, 2xy)$ is injective for y>0?

You need to show that for every $(x_1,y_1)$ and $(x_1,y_1)$ where y1 > 0 and y2 >0
if $F(x_1,y_1)=F(x_2,y_2)$ then $(x_1,y_1)=(x_2,y_2)$

$x_1^{2}-y_1^{2} = x_2^{2}-y_2^{2}$
$2x_1y_1 = 2x_2y_2$

$y_1^{2}-y_2^{2} = x_1^{2}-x_2^{2} = x_1^{2}-\frac{x_1^2y_1^2}{y_2^2} = x_1^{2}\:\left(1-\frac{y_1^2}{y_2^2}\right) = x_1^{2}\:\frac{y_2^2-y_1^2}{y_2^2}$

$(y_1^{2}-y_2^{2})\left(1+ \frac{x_1^2}{y_2^2}\right)=0$

$y_1^{2}-y_2^{2}=0$

$y_1-y_2=0$ since they are both > 0

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