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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.610/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |