I have a curve given implicitly by
F[x_,y_]=x^4 + x^3 y (-2 - 4 α) + y^4 α^2 (1 + α)^2 + d1 + y*d2 + x(-2y^3 α(1+α)(1+2α)+y(-1-2α+(1+2α)t_2^2/t_3^2 ) + y^2 ( 2(1+2α)/t_3 + (1+2α)t_2/t_3)) + x^2 ( 1 + y^2 (1+6α+6α^2) - t_2^2/t_3^2 + y(-2-t_2)/t_3 ) + y^3(-1-2α-2α^2 - α(1+α)t_2)/t_3 + y^2(1-t_2 - α(1+α)t_2^2 + α(1+α)t_3^2)/t_3^2 == 0
and I want to set the coefficients $ d_1$ and $ d_2$ such that the system of equations
$ F(x,y)=0, F_x(x,y)=0, F_y(x,y)=0$
has three solutions in $ x$ and $ y$ which will depend on $ \alpha, t_2, t_3$ .
Is there a good way of going about this without fixing $ t_2, t_3$ and $ \alpha$ ?
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