Further than Descartes' rule of signs

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.