{"title":"Real univariate polynomials with given signs of coefficients and simple real roots","authors":"V. MatematychniStudii., No 61, V. P. Kostov","doi":"10.30970/ms.61.1.22-34","DOIUrl":null,"url":null,"abstract":"We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\\ell ^+$ and $\\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\\mathcal{P}^d:=\\{ P:=x^d+a_1x^{d-1}+\\dots +a_d\\}$, $a_j\\in \\mathbb{R}^*=\\mathbb{R}\\setminus \\{ 0\\}$, the corresponding sign patterns $\\sigma=(\\sigma_1,\\sigma_2,\\dots, \\sigma_d)$, where $\\sigma_j=$sign$(a_j)$, and the sets $\\mathcal{P}^d_{\\sigma ,(\\ell ^+,\\ell ^-)}\\subset \\mathcal{P}^d$ of polynomials with given triples $(\\sigma ,(\\ell ^+,\\ell ^-))$.We prove that for degree $d\\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\\sigma =(-,-,-,+)$, $\\ell^+=0$, $\\ell ^-=2$, with $d=5$, $\\sigma =(-,-,-,-,+)$, $\\ell^+=0$, $\\ell ^-=3$, and the ones obtained from them under the $\\mathbb{Z}_2\\times \\mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\\colon P(x)\\mapsto (-1)^dP(-x)$ and $i_r\\colon P(x)\\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\\leq d$);2) for $1\\leq s\\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\\geq 6$, we give an example of a set $\\mathcal{P}^d_{\\sigma ,(\\ell^+,\\ell^-)}$ having $\\Lambda (d)$ connected compo\\-nents, where $\\Lambda (d)\\rightarrow \\infty$ as $d\\rightarrow \\infty$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.61.1.22-34","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\leq d$);2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$ having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.