Polynomials, sign patterns and Descartes' rule of signs

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $pos\le c$ positive and $neg\le p$ negative roots, where $pos\equiv c($, mod $2)$ and $neg\equiv p($mod$2)$\mathrm{.}For$1\leq d\leq 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(pos, neg)$satisfyingtheseconditionsthereexistsapolynomial$P$withexactly$pos$positiveandexactly$neg$negativeroots(allofthemsimple)\mathrm{.}For$d\geq 4$thisisnotso\mathrm{.}Itwasobservedthatfor$4\leq d\leq 10$, in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\ge 4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair.