On Descartes' rule of signs

A sequence of \(d+1\) signs \(+\) and \(-\) beginning with a \(+\) is called a {\em sign pattern (SP)}. We say that the real polynomial \(P:=x^d+\sum _{j=0}^{d-1}a_jx^j\), \(a_j\neq 0\), defines the SP \(\sigma :=(+\),sgn\((a_{d-1})\), \(\ldots\), sgn\((a_0))\). By Descartes' rule of signs, for the quantity \(pos\) of positive (resp. \(neg\) of negative) roots of \(P\), one has \(pos\leq c\) (resp. \(neg\leq p=d-c\)), where \(c\) and \(p\) are the numbers of sign changes and sign preservations in \(\sigma\); the numbers \(c-pos\) and \(p-neg\) are even. We say that \(P\) realizes the SP \(\sigma\) with the pair \((pos, neg)\). For SPs with \(c=2\), we give some sufficient conditions for the (non)realizability of pairs \((pos, neg)\) of the form \((0,d-2k)\), \(k=1\), \(\ldots\), \([(d-2)/2]\).