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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.12951 |
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| _version_ | 1866911142262079488 |
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| author | Miller, Michael J. |
| author_facet | Miller, Michael J. |
| contents | A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $β$ is one of those roots, then within one unit of $β$ lies a root of the polynomial's derivative. If we define $r(β)$ to be the greatest possible distance between $β$ and the closest root of the derivative, then Sendov's conjecture claims that $r(β) \le 1$.
In this paper, we conjecture that there is a constant $c>0$ so that $r(β) \le 1-cβ(1-β)$ for all $β\in [0,1]$. We find such constants for complex polynomials of degree $2$ and $3$, for real polynomials of degree $4$, for all polynomials whose roots lie on a line, for all polynomials with exactly one distinct critical point, and when $β$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx0.233$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12951 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Seeking a quadratic refinement of Sendov's conjecture Miller, Michael J. Complex Variables 30C15 A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $β$ is one of those roots, then within one unit of $β$ lies a root of the polynomial's derivative. If we define $r(β)$ to be the greatest possible distance between $β$ and the closest root of the derivative, then Sendov's conjecture claims that $r(β) \le 1$. In this paper, we conjecture that there is a constant $c>0$ so that $r(β) \le 1-cβ(1-β)$ for all $β\in [0,1]$. We find such constants for complex polynomials of degree $2$ and $3$, for real polynomials of degree $4$, for all polynomials whose roots lie on a line, for all polynomials with exactly one distinct critical point, and when $β$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx0.233$. |
| title | Seeking a quadratic refinement of Sendov's conjecture |
| topic | Complex Variables 30C15 |
| url | https://arxiv.org/abs/2506.12951 |