Intersections of Convex Hulls of Polynomial Shifts and Critical Points

TL;DR

This paper proves a fundamental intersection property of convex hulls of polynomial zeros and their derivatives, characterizes when these hulls coincide, and applies these results to conjectures and bounds related to polynomial critical points.

Contribution

It establishes the equality of the intersection of convex hulls of polynomial shifts with the convex hull of critical points, and provides new bounds and characterizations related to polynomial zeros and critical points.

Findings

Proves _{c \u2208 } K_c = K' for polynomial p(z).

Characterizes when K_0 = K' based on zero multiplicities.

Refines bounds on the location of critical points relative to zeros.

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 $\zeta\in K'$ the disk $|z-\zeta|\le \sqrt{1-|\zeta|^2}$ 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.