Geodesic
Polynomials, sign patterns and Descartes' rule of signs
Mathematica Bohemica, Tome 144 (2019) no. 1, pp. 39-67
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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 ${\rm pos}\leq c$ positive and $\neg \leq p$ negative roots, where ${\rm pos}\equiv c\pmod 2$ and $\neg \equiv p\pmod 2$. For $1\leq d\leq 3$, 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 $\neg $ negative roots (all of them simple). For $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\geq 4$. 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.
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 ${\rm pos}\leq c$ positive and $\neg \leq p$ negative roots, where ${\rm pos}\equiv c\pmod 2$ and $\neg \equiv p\pmod 2$. For $1\leq d\leq 3$, 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 $\neg $ negative roots (all of them simple). For $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\geq 4$. 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.
DOI : 10.21136/MB.2018.0091-17
Classification : 26C10, 30C15
Keywords: real polynomial in one variable; sign pattern; Descartes' rule of signs 
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Kostov, Vladimir Petrov. Polynomials, sign patterns and Descartes' rule of signs. Mathematica Bohemica, Tome 144 (2019) no. 1, pp. 39-67. doi: 10.21136/MB.2018.0091-17

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