Further than Descartes' rule of signs

Abstract: The {\em sign pattern} defined by the real polynomial $Q:=\Sigma _{j=0}^da_jx^j$, $a_j\neq 0$, is the string $\sigma (Q):=({\rm sgn(}a_d{\rm )},\ldots ,{\rm sgn(}a_0{\rm )})$. 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 $pos\geq 1$, $neg\geq 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern.