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

TitleA nonrealization theorem in the context of Descartes' rule of signs
Publication TypeJournal Article
Year of Publication2019
AuthorsCheriha H, Gati Y, Kostov VPetrov
JournalAnnuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique
Start Page25
ISSN1313-9215 (Print) 2603-5529 (Online)
KeywordsDescartes' rule of signs, Real polynomials, sign pattern

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 $pos\leq c$ 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 $neg$ negative roots (all of them simple); that is, all these cases are realizable. This is not true for $d\geq 4$, yet for $4\leq d\leq 8$ (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 $d\geq 4$. 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 $d=9$.

2010 MSC

Primary: 26C10; Secondary: 30C15

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