Let's say you have a polynomial depending on complex parameters $A,B$: $$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$
One parametrization of zero points of this polynomial is given by $$ x(t) = -(t + t^{-1}), \ y(t) = At + Bt^{-1}$$
This can be checked by plugging it in, but I can't figure out how you would get this parametrization only by looking at polynomial $p$. I've tried the only trick I know, finding one zero point $(x,y)$ of $p$, and looking at lines through that point but that only gave me parametrizations much more complex than this one. I suppose you could get to the simple parametrization by using the right change of coordinates, but I can't see that immediately.
Does anyone know some tricks that could be helpful here?
Put $X=y+Ax$ and $Y=y+Bx$ so that $XY=-(A-B)^2$ which is parametrised by $X=(B-A)t; Y=(A-B)t^{-1}$
Then $X-Y=(A-B)x$ and also $X-Y=(B-A)(t+t^{-1})$ hence $x=-(t+t^{-1})$
Also $BX-AY=(B-A)y$ and also $BX-AY=(B-A)(Bt+At^{-1}$)
If you use $u=t^{-1}$ you get the parametrisation you were given.