I am looking for conditions under which the only solution to the equation $Ax = x$, where the matrix $A$ is a companion matrix
\begin{equation} A = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ a_0 & a_1 & a_2 & \cdots & a_{n-1} \end{pmatrix} \end{equation}
is $x = 0$. I believe this can be proved by rewriting the equation as $(A - I) x = 0$ and investigating the null space of $A - I$ (for instance through its rank), but I am not available to find satisfying conditions, since the convenient properties of $A$ which follow from being a companion matrix seem to me to disappear when I subtract the identity matrix.
My guess would be that is should be possible to state somewhat simple conditions directly in terms of the coefficients $(a_0, \dots, a_{n-1})$, or in terms of solutions to the polynomial $p(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + x^n$ (from which the matrix $A$ originates). Either would be fine by me.