Degree $6$ hyperbolic polynomials and orders of moduli
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913536186253312 |
|---|---|
| author | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| author_facet | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| contents | We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where $\tilde{c}$ and $\tilde{p}$ are the numbers of sign changes and sign preservations in the sequence of its coefficients, $\tilde{c}+\tilde{p}=d$. For $d=6$, we give the exhaustive answer to the question: When the moduli of all $6$ roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14698 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Degree $6$ hyperbolic polynomials and orders of moduli Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki Classical Analysis and ODEs 26C10, 30C15 We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where $\tilde{c}$ and $\tilde{p}$ are the numbers of sign changes and sign preservations in the sequence of its coefficients, $\tilde{c}+\tilde{p}=d$. For $d=6$, we give the exhaustive answer to the question: When the moduli of all $6$ roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients? |
| title | Degree $6$ hyperbolic polynomials and orders of moduli |
| topic | Classical Analysis and ODEs 26C10, 30C15 |
| url | https://arxiv.org/abs/2310.14698 |