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

Abstract

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

Cited by in corpus (1)

• New aspects of Descartes' rule of signs