Numerical ranges of companion matrices

We show that an n-by-n companion matrix A can have at most n line segments on the boundary ∂W(A) of its numerical range W(A), and it has exactly n line segments on ∂W(A) if and only if, for n odd, A is unitary, and, for n even, A is unitarily equivalent to the direct sum A1 ⊕ A2 of two (n/2)-by-(n/2) companion matricesA1=010⋱⋱1a0andA2=010⋱⋱1-1/a¯0 with 1 ⩽ ∣a∣ < tan(π/n) + sec(π/n).