Polynomials, sign patterns and Descartes' rule of signs

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 ${\rm pos}\leq c$ positive and $\neg\leq p$ negative roots, where ${\rm pos}\equiv c\pmod2$ and $\neg\equiv p\pmod2$. For $1\leq d\leq3$, 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 $d\geq4$ this is not so. It was observed that for $4\leq d\leq8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\geq4$. 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.

Keywords

• real polynomial in one variable
• sign pattern
• Descartes' rule of signs