On universal sign patterns

TL;DR

This paper investigates the conditions under which certain sign patterns of polynomial coefficients are universal, meaning they can realize any order of root moduli, especially focusing on polynomials formed by products of binomials.

Contribution

It establishes that the sign pattern of polynomials like (x-1)^m(x+1)^n with non-vanishing coefficients is universal, and explores cases with vanishing coefficients.

Findings

Sign pattern of (x-1)^m(x+1)^n is universal when coefficients are non-zero.

Studied the impact of vanishing coefficients on universality.

Provided conditions for universality of sign patterns in these polynomials.

Abstract

We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$ positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these roots distinct. The {\em order} of these moduli is defined when in their string as points of the positive half-axis one marks the places of the moduli of negative roots. A sign pattern $\sigma^0$ is {\em universal} when for any possible order of the moduli there exists a polynomial $Q$ with $\sigma (Q)=\sigma^0$ and with this order of the moduli of its roots. We show that when the polynomial $P_{m,n}:=(x-1)^m(x+1)^n$ has no vanishing coefficients, the sign pattern $\sigma (P_{m,n})$ is universal. We also study the question when $P_{m,n}$ can have vanishing coefficients.