The paper discusses a distribution of the zeros of the polynomial

    \begin{equation*}p(\lambda)\equiv\lambda^{k+1}-\lambda^k+q,\qquad q\in\Bbb R,\quad k\in\Bbb Z^+ \end{equation*}

with respect to the unit circle. This problem is of theoretic as well as practical importance which motivated S. A. ~Levin and R. May to formulate a necessary and sufficient condition guaranteeing the location of all the zeros of p(\lambda ) inside the unit circle. We give a simple alternate proof of their criterion and, as the main result, present a complete list of all possible z  ero distributions of p(\lambda ) with respect to this circle.



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Čermák, Jan, Jánský, Jiří