Let $P(x,y,z)$ be a homogeneous polynomial of degree $n$. Consider the vector field $$F_P = \left[ \begin{array}{*{20}{c}} \frac{\partial P}{\partial x} - nxP\\ \frac{\partial P}{\partial y} - nyP\\ \frac{\partial P}{\partial z} - nzP \end{array} \right].$$ I want to prove that if $n$ is fixed, the number of isolated rest points of this vector field $F_P$ is uniformly bounded over all homogeneous polynomials $P$ of degree $n$. What is the bound for $n=3$?
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