Further than Descartes' rule of signs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917796607164416 |
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| author | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| author_facet | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| contents | The {\em sign pattern} defined by the real polynomial $Q:=Σ_{j=0}^da_jx^j$, $a_j\neq 0$, is the string $σ(Q):=({\rm sgn(}a_d{\rm )},\ldots ,{\rm sgn(}a_0{\rm )})$. The quantities $pos$ and $neg$ of positive and negative roots of $Q$ satisfy Descartes' rule of signs. A couple $(σ_0,(pos,neg))$, where $σ_0$ is a sign pattern of length $d+1$, is {\em realizable} if there exists a polynomial $Q$ with $pos$ positive and $neg$ negative simple roots, with $(d-pos-neg)/2$ complex conjugate pairs and with $σ(Q)=σ_0$. We present a series of couples (sign pattern, pair $(pos,neg)$) depending on two integer parameters and with $pos\geq 1$, $neg\geq 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Further than Descartes' rule of signs Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki Classical Analysis and ODEs The {\em sign pattern} defined by the real polynomial $Q:=Σ_{j=0}^da_jx^j$, $a_j\neq 0$, is the string $σ(Q):=({\rm sgn(}a_d{\rm )},\ldots ,{\rm sgn(}a_0{\rm )})$. The quantities $pos$ and $neg$ of positive and negative roots of $Q$ satisfy Descartes' rule of signs. A couple $(σ_0,(pos,neg))$, where $σ_0$ is a sign pattern of length $d+1$, is {\em realizable} if there exists a polynomial $Q$ with $pos$ positive and $neg$ negative simple roots, with $(d-pos-neg)/2$ complex conjugate pairs and with $σ(Q)=σ_0$. We present a series of couples (sign pattern, pair $(pos,neg)$) depending on two integer parameters and with $pos\geq 1$, $neg\geq 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern. |
| title | Further than Descartes' rule of signs |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2302.04540 |