I am given the the function $F(t, x, y) := \left( \begin{array}{} 3x^3 +2y^5-5t^7 \\ 2x^3-y^5-t^7\\ \end{array}\right) = \left( \begin{array}{} 0 \\ 0\\ \end{array}\right)$ where I should show that it has a locally distinct solution at $(t,x,y)=(0,0,0) $ for $x,y$ even though the implicit function theorem does not work. Could you give me a hint as to how I can tackle this given that I know that $det(\frac{\partial F}{\partial x\partial y})=0$ at $(0,0,0)$?
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1Are you sure that there's other solution for the system $3x^3+2y^5-5t^7=0; 2x^3-y^5-t^7=0$? I mean, if you multiply the second equation by a factor of $2$ and you add up, you will get the equation $x^3=t^7$, right? Now you have two new equations that depends only on $x$ and $y$, whose solutions seems to be only the $(0,0)$ – rowcol Apr 09 '21 at 19:41
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Thank you. I was so caught up with the IFT, I forgot about basic algebra. – Alex Apr 09 '21 at 19:50
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1In general, when the implicit function fails, maybe you could try to find solutionts by other (maybe algebraic) methods – rowcol Apr 09 '21 at 19:55