Further than Descartes’ rule of signs
The sign pattern defined by the real polynomial $Q:=\Sigma _{j=0}^da_jx^j$, $a_j\ne 0$, is the string $\sigma (Q):=(\mathrm{sgn}(a_d),\ldots ,\mathrm{sgn}(a_0))$. The quantities $\mathrm{pos}$ and $\mathrm{neg}$ of positive and negative roots of $Q$ satisfy Descartes’ rule of signs. A couple $(\sigm...
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.610/ |
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| author | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| author_facet | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki |
| author_sort | Gati, Yousra |
| collection | DOAJ |
| description | The sign pattern defined by the real polynomial $Q:=\Sigma _{j=0}^da_jx^j$, $a_j\ne 0$, is the string $\sigma (Q):=(\mathrm{sgn}(a_d),\ldots ,\mathrm{sgn}(a_0))$. The quantities $\mathrm{pos}$ and $\mathrm{neg}$ of positive and negative roots of $Q$ satisfy Descartes’ rule of signs. A couple $(\sigma _0,(\mathrm{pos},\mathrm{neg}))$, where $\sigma _0$ is a sign pattern of length $d+1$, is realizable if there exists a polynomial $Q$ with $\mathrm{pos}$ positive and $\mathrm{neg}$ negative simple roots, with $(d-\mathrm{pos}-\mathrm{neg})/2$ complex conjugate pairs and with $\sigma (Q)=\sigma _0$. We present a series of couples (sign pattern, pair $(\mathrm{pos},\mathrm{neg})$) depending on two integer parameters and with $\mathrm{pos}\ge 1$, $\mathrm{neg}\ge 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern. |
| format | Article |
| id | doaj-art-85fa70d0c8e4420aa38fcb889feeb57d |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-85fa70d0c8e4420aa38fcb889feeb57d2025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G886388110.5802/crmath.61010.5802/crmath.610Further than Descartes’ rule of signsGati, Yousra0Kostov, Vladimir Petrov1Tarchi, Mohamed Chaouki2Université de Carthage, EPT-LIM, TunisieUniversité Côte d’Azur, CNRS, LJAD, FranceUniversité de Carthage, EPT-LIM, TunisieThe sign pattern defined by the real polynomial $Q:=\Sigma _{j=0}^da_jx^j$, $a_j\ne 0$, is the string $\sigma (Q):=(\mathrm{sgn}(a_d),\ldots ,\mathrm{sgn}(a_0))$. The quantities $\mathrm{pos}$ and $\mathrm{neg}$ of positive and negative roots of $Q$ satisfy Descartes’ rule of signs. A couple $(\sigma _0,(\mathrm{pos},\mathrm{neg}))$, where $\sigma _0$ is a sign pattern of length $d+1$, is realizable if there exists a polynomial $Q$ with $\mathrm{pos}$ positive and $\mathrm{neg}$ negative simple roots, with $(d-\mathrm{pos}-\mathrm{neg})/2$ complex conjugate pairs and with $\sigma (Q)=\sigma _0$. We present a series of couples (sign pattern, pair $(\mathrm{pos},\mathrm{neg})$) depending on two integer parameters and with $\mathrm{pos}\ge 1$, $\mathrm{neg}\ge 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.610/Real polynomial in one variablehyperbolic polynomialsign patternDescartes’ rule of signs |
| spellingShingle | Gati, Yousra Kostov, Vladimir Petrov Tarchi, Mohamed Chaouki Further than Descartes’ rule of signs Comptes Rendus. Mathématique Real polynomial in one variable hyperbolic polynomial sign pattern Descartes’ rule of signs |
| title | Further than Descartes’ rule of signs |
| title_full | Further than Descartes’ rule of signs |
| title_fullStr | Further than Descartes’ rule of signs |
| title_full_unstemmed | Further than Descartes’ rule of signs |
| title_short | Further than Descartes’ rule of signs |
| title_sort | further than descartes rule of signs |
| topic | Real polynomial in one variable hyperbolic polynomial sign pattern Descartes’ rule of signs |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.610/ |
| work_keys_str_mv | AT gatiyousra furtherthandescartesruleofsigns AT kostovvladimirpetrov furtherthandescartesruleofsigns AT tarchimohamedchaouki furtherthandescartesruleofsigns |