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After formulating a problem I found that the solution should be the one that is given by the pair of equations \begin{align*} 990 x^7 - 112 x^8 + 165 y^7 - 35 x y^7 + 21 x^3 y^4 (-25 + 7 y) = 0, \\ 75 x^7 - 35 x^7 y + 21 x^4 y^3(-55 + 7 x) + 450 y^7 - 112 y^8 = 0, \end{align*} However, I'm having trouble to see that this pair really has an unique solution in $\mathbb{R}_+^2 \setminus (0,0)$ (as it is most probably impossible to solve it explicitly). Here $\mathbb{R}_+^2 \setminus (0,0)$ means that both $x$ and $y$ should be strictly positive. According to numerical calculations it does, but how can I prove it?

I'm very interested if someone can come up with a method that also includes cases where the two equations do not only include polynomials.

To be more precise, I bet doing this will be very cumbersome and I'm more interested in ideas how to proceed and tackle problems like this than solving this one explicit case! Any references and ideas are welcome.

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  • $\begingroup$ I'm very interested in the source of the problem. $\endgroup$ – Will Jagy Jun 13 '20 at 17:55
  • $\begingroup$ @WillJagy Well it is not a short story. Hopefully, the solution to this pair of equations characterizes the unique optimal control to a quite specific control problem where underlying dynamics are determined by geometric Brownian motion. In reality the pair is much more complicated and this is just an example what the output looks like but I'm hoping that the solution could be generalized. I'm just completely out of ideas how to even start proving this kind of thing. $\endgroup$ – NPHA Jun 13 '20 at 21:37
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Alright. There is an evident solution at about $(9,2).$ Then have curves have asymptotes near $y=x$ in the first quadrant. It is fairly likely that these asymptotes do not intersect, and this can be investigated more carefully. We usually think in terms of slope and $y-x,$ so let $x = -u+v$ and $y = u+v,$ so that $y-x = 2u$ and $y+x = 2v.$ Both curves have arcs with large positive $v$ and small $u,$ I think slightly negative. So: more work is needed, but it can be done.

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  • $\begingroup$ Can you elaborate a bit more on your idea as I’m not really following. What do you mean by ” Both curves have arcs with large positive $v$ and small $u$, I think slightly negative. ”? $\endgroup$ – NPHA Jun 14 '20 at 20:11

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