Three realization problems about univariate polynomials

Abstract: We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $σ(P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$. The numbers $p_d$ and $n_d$ of positive and negative roots of $P$ (counted with multiplicity) satisfy the Descartes' rule of signs. Problem~1 asks for which couples $C$ of the form (sign pattern $σ$, pair $(p_d,n_d)$ compatible with $σ$ in the sense of Descartes' rule of signs), there exist polynomials $P$ defining these couples. Problem~2 asks for which $d$-tuples of pairs $T:=((p_d,n_d)$, $\ldots$, $(p_1,n_1))$, there exist polynomials $P$ such that $P^{(d-j)}$ has $p_j$ positive and $n_j$ negative roots. A $d$-tuple $T$ determines the sign pattern $σ(P)$, but the inverse is false. We show by an example that $6$ is the smallest value of $d$ for which there exist non-realizable tuples $T$ for which the corresponding couples $C$ are realizable. The third problem concerns polynomials with all roots real. We give a geometric interpretation of the three problems in the context of degree $4$ polynomials.