Descartes' rule of signs, Rolle's theorem and sequences of admissible pairs

TL;DR

This paper characterizes the sequences of positive and negative roots of a polynomial and its derivatives, satisfying Rolle's theorem and Descartes' rule of signs, for degrees up to five, with all roots simple and coefficients nonvanishing.

Contribution

It provides a complete characterization of admissible root count sequences for polynomials of degree up to five with nonvanishing coefficients, extending classical root sign rules.

Findings

For degrees 1 to 5, identified which admissible root count sequences are realizable.

Established existence of polynomials with prescribed root counts and simple roots.

Connected root count sequences with classical inequalities from Rolle's theorem and Descartes' rule.

Abstract

Given a real univariate degree $d$ polynomial $P$, the numbers $pos_k$ and $neg_k$ of positive and negative roots of $P^{(k)}$, $k=0$, $\ldots$, $d-1$, must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's theorem and from Descartes' rule of signs. For $1\leq d\leq 5$, we give the answer to the question for which admissible $d$-tuples of pairs $(pos_k$, $neg_k)$ there exist polynomials $P$ with all nonvanishing coefficients such that for $k=0$, $\ldots$, $d-1$, $P^{(k)}$ has exactly $pos_k$ positive and $neg_k$ negative roots all of which are simple.