A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $\mathbb{R}$.
Assume that $p(x,y)\in \mathbb{C}[x,y]$ has all its roots in a real half space $K$. Is it true that all critical points of $p$ must necessarily lie in $K$?



A possible generalization of Gauss Lucas theorem to higher dimension
There is a very pretty multivariate extension of Gauss-Lucas proved by Marek Kanter [here](https://arxiv.org/abs/1203.6426) As far as I can tell, the paper has not been published, but the proof of the main theorem is half a page, so... 


