Polynomials, sign patterns and Descartes' rule of signs

TL;DR

This paper investigates the realizability of root sign patterns in polynomials based on Descartes' rule of signs, providing new counterexamples for degree 11 that challenge previous conjectures about root distributions.

Contribution

The authors construct a counterexample for degree 11, disproving the conjecture that certain root sign patterns are always realizable for degrees greater than or equal to 4.

Findings

Counterexample for degree 11 with specific sign pattern and root counts

Disproof of the conjecture that all nonrealizable cases have zero positive or negative roots

Extension of understanding of root sign pattern realizability in polynomials

Abstract

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 $pos\leq c$ positive and $neg\leq p$ negative roots, where $pos\equiv c($\, mod $2)$ and $neg\equiv p($\, mod $2)$. 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 $(pos, neg)$ satisfying these conditions there exists a polynomial $P$ with exactly $pos$ positive and exactly $neg$ negative roots (all of them simple). For $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 10$, in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\geq 4$. 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.