Intersections of Convex Hulls of Polynomial Shifts and Critical Points
Teng Zhang
TL;DR
The paper addresses locating the critical points of a complex polynomial by studying the convex-hull geometry of shifted zeros, proving the key identity $\bigcap_{c\in\mathbb{C}} K_c = K'$. It combines a separating-hyperplane argument with a half-plane non-surjectivity theorem for polynomials without critical points, with the latter established via analytic continuation, the Monodromy Theorem, and Liouville’s theorem. It yields a precise characterization for when $K_0=K'$, namely that every vertex of $K_0$ is a multiple zero of $p$, and derives Schmeisser-type disk containment results under unit-disk zeros, along with a refined barycentric bound placing a critical point within $\sqrt{\frac{n-2}{n-1}}\sqrt{1-|G|^2}$ of the zeros’ centroid $G$. These results advance convex-analytic methods in critical-point problems, providing sharper quantitative bounds and connecting to classical conjectures in polynomial location.
Abstract
Let $p(z)$ be a complex polynomial of degree $n\ge 2$. For each $c\in\mathbb{C}$, let $K_c$ denote the convex hull of the zeros of $p(z)+c$, and let $K'$ denote the convex hull of the zeros of $p'(z)$. We prove that $$\bigcap_{c\in\mathbb{C}} K_c = K',$$ by combining a strict separating hyperplane argument with a half-plane non-surjectivity theorem for polynomials without critical points (proved via analytic continuation, the monodromy theorem and Liouville's Theorem). We also characterize when $K_0=K'$ in terms of the multiplicities of the zeros of $p(z)$ that form the vertices of $K_0$. As an application, we obtain a partial result toward the Schmeisser's conjecture: if all zeros of $p$ lie in the closed unit disk, then for every $ζ\in K'$ the disk $|z-ζ|\le \sqrt{1-|ζ|^2}$ contains a critical point of $p(z)$. Finally, we refine a recent barycentric bound in \cite{Zha26+} by showing that there is always a critical point within distance $\sqrt{\frac{n-2}{n-1}}\sqrt{1-|G|^2}$ of the centroid $G$ of the zeros.