Further than Descartes' rule of signs

Abstract

The {\em sign pattern} defined by the real polynomial $Q:={\Sigma }_{j=0}^d{a}_j{x}^j$, $a_j\ne 0$, is the string $\sigma (Q):=(\mathrm{sgn}(a_d),\dots ,\mathrm{sgn}(a_0))$. The quantities $pos$ and $neg$ of positive and negative roots of $Q$ satisfy Descartes' rule of signs. A couple $({\sigma }_0,(pos,neg))$, where ${\sigma }_0$ is a sign pattern of length $d+1$, is {\em realizable} if there exists a polynomial $Q$ with $pos$ positive and $neg$ negative simple roots, with $(d-pos-neg)/2$ complex conjugate pairs and with $\sigma (Q)={\sigma }_0$. We present a series of couples (sign pattern, pair $(pos,neg)$) depending on two integer parameters and with $pos\ge 1$, $neg\ge 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern.