Three realization problems about univariate polynomials

TL;DR

The paper investigates three realization problems for monic real univariate polynomials: realizability of compatible sign-pattern pairs, realizability of sequences of root counts across derivatives (SCPs), and a real-rooted-polynomial variant linked to discriminant geometry. It develops a discriminant/hyperbolic-polynomial framework to connect sign patterns, root distributions, and derivative behavior, deriving complete results for small degrees and producing a degree-6 counterexample showing SCP non-realizability independent of compatible-couple non-realizability. The work reveals rich geometric structures in coefficient spaces and uses degree-4 illustrations to ground abstract results in discriminant manifolds and boundary phenomena, highlighting how local root configurations constrain global sign-pattern realizability.

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.

Figures (1)

• Figure 1: The orthants $\mathcal{R}$ and $\mathcal{R}^{\dagger}$.

Theorems & Definitions (24)

• Definition 1
• Example 1
• Remark 1
• Definition 2
• Remark 2
• Remark 3
• Remark 4
• Proposition 1
• Theorem 1
• Definition 3