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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 $K$?

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    $\begingroup$ Can you provide an example? $\endgroup$ – geocalc33 Mar 3 at 2:25
  • $\begingroup$ @geocalc33 As trivial example $z=0$.There is no critical point. $\endgroup$ – Ali Taghavi Mar 3 at 2:28
  • $\begingroup$ The real half space is $Re(z)>-1$ $\endgroup$ – Ali Taghavi Mar 3 at 2:35
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    $\begingroup$ You might be able to use a convex space argument here because the real half space is a convex set. Given a set $X,$ a convexity over $X$ is a collection $𝒞$ of subsets of X satisfying the $(X, 𝒞).$ It's hard to refine the convex sets in question in higher dimensions. I would suggest reading this paper, which delves into generalisations of Gauss Lucas: researchgate.net/publication/221966131_Multivariate_Gauss-Lucas_Theorems $\endgroup$ – geocalc33 Mar 3 at 3:38
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    $\begingroup$ and this: [4] A. W. Goodman: Remarks on the Gauss-Lucas theorem in higher dimensional space,Proc. Amer. Math. Soc. 55 (1976), 97–12 $\endgroup$ – geocalc33 Mar 3 at 3:39
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There is a very pretty multivariate extension of Gauss-Lucas proved by Marek Kanter here As far as I can tell, the paper has not been published, but the proof of the main theorem is half a page, so...

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