A nonrealization theorem in the context of Descartes' rule of signs

Abstract

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\le c$ positive and $neg\le p$ negative roots, where $pos\equiv c($ mod $2)$ and $neg\equiv p($ mod $2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(pos,neg)$ satisfying these conditions there exists a polynomial $P$ with exactly $pos$ positive and $neg$ negative roots (all of them simple); that is, all these cases are realizable. This is not true for $d\ge 4$, yet for $4\le d\le 8$ (for these degrees the exhaustive answer to the question of realizability is known) in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\ge 4$. For $d=9$, we show a counterexample to this conjecture: for the sign pattern $(+,-,-,-,-,+,+,+,+,-)$ and the pair $(1,6)$ there exists no polynomial with $1$ positive, $6$ negative simple roots and a complex conjugate pair and, up to equivalence, this is the only case for $d=9$.