A refinement of Pawlowski's result
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911096992956416 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | Let \(F(z) = \prod_{k=1}^{n}(z - z_k)\) be a monic complex polynomial of degree \(n\) whose zeros satisfy \(\max\limits_{1 \le k \le n} |z_k| \le 1\). Pawłowski [Trans. Amer. Math. Soc. 350(11) (1998)] considered the radius \(γ_n\) of the smallest disk, centered at the centroid \(\frac{1}{n}\sum_{k=1}^n z_k\), containing at least one critical point of \(F\), establishing the bound $γ_n \le \frac{2\,n^{\frac{1}{n-1}}}{n^{\frac{2}{n-1}} + 1}$. In this paper, inspired by the spirit of Borcea's variance conjectures and leveraging the classical Schoenberg inequality, we significantly refine Pawłowski's estimate by proving succinctly and elegantly that $γ_n \le \sqrt{\frac{n - 2}{n - 1}}$. This result also represents a rare and noteworthy application of Schoenberg's inequality to the geometry of polynomial critical points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07105 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A refinement of Pawlowski's result Zhang, Teng Complex Variables 30C10 Let \(F(z) = \prod_{k=1}^{n}(z - z_k)\) be a monic complex polynomial of degree \(n\) whose zeros satisfy \(\max\limits_{1 \le k \le n} |z_k| \le 1\). Pawłowski [Trans. Amer. Math. Soc. 350(11) (1998)] considered the radius \(γ_n\) of the smallest disk, centered at the centroid \(\frac{1}{n}\sum_{k=1}^n z_k\), containing at least one critical point of \(F\), establishing the bound $γ_n \le \frac{2\,n^{\frac{1}{n-1}}}{n^{\frac{2}{n-1}} + 1}$. In this paper, inspired by the spirit of Borcea's variance conjectures and leveraging the classical Schoenberg inequality, we significantly refine Pawłowski's estimate by proving succinctly and elegantly that $γ_n \le \sqrt{\frac{n - 2}{n - 1}}$. This result also represents a rare and noteworthy application of Schoenberg's inequality to the geometry of polynomial critical points. |
| title | A refinement of Pawlowski's result |
| topic | Complex Variables 30C10 |
| url | https://arxiv.org/abs/2411.07105 |