On Descartes' rule of signs

Abstract

A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+{\sum }_{j=0}^{d-1}{a}_j{x}^j$, $a_j\ne 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$, $\dots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of positive (resp. $neg$ of negative) roots of $P$, one has $pos\le c$ (resp. $neg\le p=d-c$), 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 $(0,d-2k)$, $k=1$, $\dots$, $[(d-2)/2]$.