Midpoints and critical points
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917132795641856 |
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| author | Gati, Yousra Kostov, Vladimir Petrov |
| author_facet | Gati, Yousra Kostov, Vladimir Petrov |
| contents | For a degree $5$ real polynomial with roots $x_1\leq \cdots \leq x_5$ and roots $ξ_1\leq \cdots \leq ξ_4$ of its derivative, we set $z_j:=(x_j+x_{j+1})/2$, $1\leq j\leq 4$. We prove that one cannot have at the same time $\min_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \min_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$ and $\max_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \max_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$. The result settles a general question about midpoints and critical points of hyperbolic polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Midpoints and critical points Gati, Yousra Kostov, Vladimir Petrov Classical Analysis and ODEs For a degree $5$ real polynomial with roots $x_1\leq \cdots \leq x_5$ and roots $ξ_1\leq \cdots \leq ξ_4$ of its derivative, we set $z_j:=(x_j+x_{j+1})/2$, $1\leq j\leq 4$. We prove that one cannot have at the same time $\min_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \min_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$ and $\max_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \max_{1\leq j\leq 3}(ξ_{j+1}-ξ_j)$. The result settles a general question about midpoints and critical points of hyperbolic polynomials. |
| title | Midpoints and critical points |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2512.07322 |