Polynomials, sign patterns and Descartes' rule of signs

Kostov, Vladimir Petrov. "Polynomials, sign patterns and Descartes' rule of signs." Mathematica Bohemica 144.1 (2019): 39-67. <http://eudml.org/doc/294764>.

@article{Kostov2019,
abstract = {By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\{\rm pos\}\le c$ positive and $\lnot \le p$ negative roots, where $\{\rm pos\}\equiv c\hspace\{4.44443pt\}(\@mod \; 2)$ and $\lnot \equiv p\hspace\{4.44443pt\}(\@mod \; 2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\{\rm pos\}, \{\rm neg\})$ satisfying these conditions there exists a polynomial $P$ with exactly $\{\rm pos\}$ positive and exactly $\lnot $ negative roots (all of them simple). For $d\ge 4$ this is not so. It was observed that for $4\le d\le 8$, in all nonrealizable cases either $\{\rm pos\}=0$ or $\{\rm neg\}=0$. It was conjectured that this is the case for any $d\ge 4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair.},
author = {Kostov, Vladimir Petrov},
journal = {Mathematica Bohemica},
keywords = {real polynomial in one variable; sign pattern; Descartes' rule of signs},
language = {eng},
number = {1},
pages = {39-67},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Polynomials, sign patterns and Descartes' rule of signs},
url = {http://eudml.org/doc/294764},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Kostov, Vladimir Petrov
TI - Polynomials, sign patterns and Descartes' rule of signs
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 1
SP - 39
EP - 67
AB - By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has ${\rm pos}\le c$ positive and $\lnot \le p$ negative roots, where ${\rm pos}\equiv c\hspace{4.44443pt}(\@mod \; 2)$ and $\lnot \equiv p\hspace{4.44443pt}(\@mod \; 2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $({\rm pos}, {\rm neg})$ satisfying these conditions there exists a polynomial $P$ with exactly ${\rm pos}$ positive and exactly $\lnot $ negative roots (all of them simple). For $d\ge 4$ this is not so. It was observed that for $4\le d\le 8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\ge 4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair.
LA - eng
KW - real polynomial in one variable; sign pattern; Descartes' rule of signs
UR - http://eudml.org/doc/294764
ER -