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

Publication Type | Journal Article |

Year of Publication | 2019 |

Authors | Cheriha H, Gati Y, Kostov VPetrov |

Journal | Annuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique |

Volume | 106 |

Start Page | 25 |

Pagination | 25-51 |

ISSN | 1313-9215 (Print) 2603-5529 (Online) |

Keywords | Descartes' rule of signs, Real polynomials, sign pattern |

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 $pos\leq c$ 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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