# intersections of two algebraic curves

Consider the following two algebraic curves on $\Bbb{R}^2$. $x$ and $y$ are variables. \begin{align*} \left(x^2 p_{-1,-1} + x p_{0,-1} + p_{1,-1}\right) y^2 + \left(x^2 p_{-1,0} + x p_{0,0} + p_{1,0} - x\right) y + \left( x^2 p_{-1,1} + x p_{-1,0} + p_{1,1}\right) &= 0\\ \left(x^2 p_{-1,-1} + x p_{0,-1} + p_{1,-1}\right) y + x^2 h_{-1} + x h_0 + h_{1} &= x \end{align*} where $$p_{0,0} + p_{1,0} + p_{1,1} + p_{0,1} + p_{-1,1} + p_{-1,0} + p_{-1,-1} + p_{0,-1} + p_{1,-1} = 1$$ and $$p_{1,1} + p_{0,1} + p_{-1,1} + h_{1} + h_{0} + h_{-1} = 1.$$ All the parameters are non-negative.

I know that the point $\left(1, \left(p_{1,1} + p_{0,1} + p_{-1,1}\right)/\left(p_{-1,-1} + p_{0,-1} + p_{1,-1}\right)\right)$ is an intersection of these two curves. I have plotted these two curves in Matlab with difference choices of parameters. Now I conjecture that these two curves have precisely two intersections when x and y are non-negative (in the first quadrant). Any idea which approaches I could start with? Thanks.

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