Three coefficients of a polynomial can determine its ϕ-instability

Let . A polynomial with real positive coefficients is said to be ϕ-stable if any root reiθ of P(x) satisfies that r > 0 and θ ∈ (ϕ, 2π − ϕ). We will see that in certain cases it is enough to know three coefficients of P(x) in order to conclude that P(x) is ϕ-unstable.The case was considered in [A. Borobia, S. Dormido, Three coefficients of a polynomial can determine its instability, Linear Algebra Appl. 338 (2001) 67–76] (note that -stability is Hurwitz stability). Now assume that , that k is an integer with 0 < k < n and that we know the coefficients a0, ak and an of P(x). We will calculate a positive number γ=γ(ϕ, n, k, a0, an) with the following property: if ak ⩽ γ then P(x) is ϕ-unstable, and if ak > γ then P(x) is ϕ-stable or ϕ-unstable depending on the rest of its coefficients.