Gauss--Lucas theorem (Tag T-C7GBB)
Let \(p\in\CC[z]\) be a nonconstant polynomial. Then every zero of \(p'\) lies in the convex hull of the set of zeros of \(p\).
Write \(p(z)=a\prod_{j=1}^n(z-a_j)\) with \(a\neq0\), \(n\ge1\), and \(a_1,\ldots,a_n\in\CC\) the zeros of \(p\) listed with multiplicity. Let \(z\in\CC\) with \(p'(z)=0\). If \(p(z)=0\), then \(z\) is one of the \(a_j\) and lies in the convex hull. Otherwise \(z\neq a_j\) for every \(j\), and \[ 0=\frac{p'(z)}{p(z)}=\sum_{j=1}^n\frac{1}{z-a_j}=\sum_{j=1}^n\frac{\overline{z}-\overline{a_j}}{\abs{z-a_j}^2}. \] Rearranging gives \[ \Bigl(\sum_{j=1}^n\frac{1}{\abs{z-a_j}^2}\Bigr)\overline{z}=\sum_{j=1}^n\frac{1}{\abs{z-a_j}^2}\overline{a_j}. \] Taking complex conjugates and dividing by \(\sum_{j}\abs{z-a_j}^{-2}>0\) writes \(z=\sum_j t_ja_j\) with \(t_j=\abs{z-a_j}^{-2}/\sum_k\abs{z-a_k}^{-2}\). The weights \(t_j\) are positive and sum to \(1\), so \(z\) lies in the convex hull of \(\{a_1,\ldots,a_n\}\).
(Gauss--Lucas)
If \(f\) is a nonconstant polynomial with zeros \(a_1, \ldots, a_n\), listed with multiplicity, then every zero of \(f'\) lies in the convex hull \(\operatorname{conv}\ts{a_1, \ldots, a_n}\).
Let \(f'(w) = 0\) and suppose \(w \notin \operatorname{conv}\ts{a_1, \ldots, a_n}\); in particular \(f(w)\neq 0\). A point outside a compact convex set is strictly separated from it by a line, so after a rotation \(z\mapsto e^{i\phi}z\) of the plane, which changes neither the hypotheses nor the conclusion, \(\Re(a_k) < \Re(w)\) for every \(k\). Writing \(f(z) = c\prod_{k=1}^n (z-a_k)\), \[ \frac{f'(w)}{f(w)} = \sum_{k=1}^n \frac{1}{w - a_k}, \] and \(\Re\qty{\frac{1}{w-a_k}} = \frac{\Re(w-a_k)}{\abs{w-a_k}^2} > 0\) for each \(k\). So \(\Re\qty{f'(w)/f(w)} > 0\), contradicting \(f'(w) = 0\).
The zeros of \(f'\) can lie on the boundary of the convex hull or in its interior. For \(f(z) = z^n\) the hull is \(\ts{0}\), and \(f'(z) = nz^{n-1}\) vanishes only at \(0\). For \(f(z) = z^3 - 1\) the hull is the triangle whose vertices are the cube roots of unity, and \(f'(z) = 3z^2\) vanishes only at the interior point \(0\).
If all zeros of a nonconstant polynomial \(f\) lie in a closed half-plane \(H\), then all zeros of \(f^{(k)}\) lie in \(H\) for every \(k\) such that \(f^{(k)}\) is nonconstant.
\(H\) is convex, so it contains the convex hull of the zeros of \(f\), and the theorem gives the case \(k=1\); induct on \(k\).