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 $(0,0,0)$?
2026-03-29 19:08:28.1774811308
Showing that a function has a distinct solution (IFT)
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