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Given the function

\begin{equation} f(x,y,z) = (x y - a_1 )^2 + [(x - a_2)^2 +2 a_3 (x - a_2)(y - a_4) + (y - a_4)^2 + a_5 z^2 ] . \end{equation}

Is there a method of finding the maximum number of stable stationary points for all $a_i$, i.e. the maximum number of points that satisfy

\begin{equation} \dfrac{\partial f}{\partial x} = 0 \quad,\quad \dfrac{\partial f}{\partial y} = 0 \quad,\quad \dfrac{\partial f}{\partial z} = 0 \quad,\quad \text{Hessian}(f)>0 \quad,\quad x, y, z, a_i \in \mathbb{R} \end{equation}

for any set of parameters $a_i$

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