On Circular Numerical Ranges of Companion Matrices with Repeated Eigenvalues

We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $\sigma(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue $a$ and the radius parameter $\lambda$, so direct coefficient comparison does not isolate $a$. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the circularity condition gives $a=0$. Furthermore, we extend this result when the spectrum is $\sigma(A) = \{0, a\}$ with algebraic multiplicities $n-m$ and $m$, respectively.