On universal sign patterns (2405.18895v1)

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.