# A possible generalization of Gauss Lucas theorem to higher dimension

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 $$K$$?

• Can you provide an example? – geocalc33 Mar 3 at 2:25
• @geocalc33 As trivial example $z=0$.There is no critical point. – Ali Taghavi Mar 3 at 2:28
• The real half space is $Re(z)>-1$ – Ali Taghavi Mar 3 at 2:35
• 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 – geocalc33 Mar 3 at 3:38
• and this: [4] A. W. Goodman: Remarks on the Gauss-Lucas theorem in higher dimensional space,Proc. Amer. Math. Soc. 55 (1976), 97–12 – geocalc33 Mar 3 at 3:39