Intersections of Convex Hulls of Polynomial Shifts and Critical Points
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914351543222272 |
|---|---|
| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intersections of Convex Hulls of Polynomial Shifts and Critical Points Zhang, Teng Complex Variables 30C10, 30C15, 52A10 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. |
| title | Intersections of Convex Hulls of Polynomial Shifts and Critical Points |
| topic | Complex Variables 30C10, 30C15, 52A10 |
| url | https://arxiv.org/abs/2601.16102 |