On Descartes' rule of signs
On Descartes' rule for polynomials with two variations of sign, Lith. Math. J. 60 (2020), no. 4, 456-469 A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum _{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP...
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| creator | Cheriha, Hassen Gati, Yousra Kostov, Vladimir Petrov |
| description | On Descartes' rule for polynomials with two variations of sign,
Lith. Math. J. 60 (2020), no. 4, 456-469 A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em
sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum
_{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$,
$\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of
positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp.
$neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign
preservations in $\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say
that $P$ realizes the SP $\sigma$ with the pair $(pos, neg)$. For SPs with
$c=2$, we give some sufficient conditions for the (non)realizability of pairs
$(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$. |
| doi_str_mv | 10.48550/arxiv.1905.01836 |
| format | Article |
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Lith. Math. J. 60 (2020), no. 4, 456-469 A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em
sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum
_{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$,
$\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of
positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp.
$neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign
preservations in $\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say
that $P$ realizes the SP $\sigma$ with the pair $(pos, neg)$. For SPs with
$c=2$, we give some sufficient conditions for the (non)realizability of pairs
$(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$.</description><identifier>DOI: 10.48550/arxiv.1905.01836</identifier><language>eng</language><subject>Mathematics - Classical Analysis and ODEs</subject><creationdate>2019-05</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1905.01836$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1905.01836$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Cheriha, Hassen</creatorcontrib><creatorcontrib>Gati, Yousra</creatorcontrib><creatorcontrib>Kostov, Vladimir Petrov</creatorcontrib><title>On Descartes' rule of signs</title><description>On Descartes' rule for polynomials with two variations of sign,
Lith. Math. J. 60 (2020), no. 4, 456-469 A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em
sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum
_{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$,
$\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of
positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp.
$neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign
preservations in $\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say
that $P$ realizes the SP $\sigma$ with the pair $(pos, neg)$. For SPs with
$c=2$, we give some sufficient conditions for the (non)realizability of pairs
$(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$.</description><subject>Mathematics - Classical Analysis and ODEs</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzrkKwkAUQNFpLCT6AWJhOqvEebNlUoo7BNKkD8_kjQTcmFHRvxeX6naXw9gIeKqs1nyG_tk9Usi5TjlYafpsXJ7jJYUG_Y3CNPb3I8UXF4fucA4D1nN4DDT8N2LVelUttklRbnaLeZGgyUwCBBIRnLXcQUZNroWTguS-ddwIUg50S7m1QglEScoorgw0AjIrwalWRmzy23519dV3J_Sv-qOsv0r5BtiZNPA</recordid><startdate>20190506</startdate><enddate>20190506</enddate><creator>Cheriha, Hassen</creator><creator>Gati, Yousra</creator><creator>Kostov, Vladimir Petrov</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20190506</creationdate><title>On Descartes' rule of signs</title><author>Cheriha, Hassen ; Gati, Yousra ; Kostov, Vladimir Petrov</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a676-1e13aa1f880f17ec952f32e3bdf062e4f15de988242aa3e4640461c217831f4d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Mathematics - Classical Analysis and ODEs</topic><toplevel>online_resources</toplevel><creatorcontrib>Cheriha, Hassen</creatorcontrib><creatorcontrib>Gati, Yousra</creatorcontrib><creatorcontrib>Kostov, Vladimir Petrov</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Cheriha, Hassen</au><au>Gati, Yousra</au><au>Kostov, Vladimir Petrov</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On Descartes' rule of signs</atitle><date>2019-05-06</date><risdate>2019</risdate><abstract>On Descartes' rule for polynomials with two variations of sign,
Lith. Math. J. 60 (2020), no. 4, 456-469 A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em
sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum
_{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$,
$\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of
positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp.
$neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign
preservations in $\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say
that $P$ realizes the SP $\sigma$ with the pair $(pos, neg)$. For SPs with
$c=2$, we give some sufficient conditions for the (non)realizability of pairs
$(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$.</abstract><doi>10.48550/arxiv.1905.01836</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Classical Analysis and ODEs |
| title | On Descartes' rule of signs |
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