Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A ⊕ B, where A ∈ M_{n - 2}, B ∈ M₂ with σ (B) = {a ω₁, a ω₂}, ω₁ⁿ = ω₂ⁿ = 1, ω₁ ≠ ω₂, and | a | ≥ fenced(| ω₁ + ω₂ | + sqrt(| ω₁ + ω₂ |² + 4 (1 + 2 cos (π / n)))) / 2.