Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices (Q802702)

A polygon in the complex plane is called cyclic if it is the convex hull of 1, \(\lambda\), \(\lambda^ 2,...\), for some complex number \(\lambda\). The regions where \(\lambda\) should lie in order that this polygon be n- sided are determined. As an application of this result the region in the complex plane in which the roots of the equation \(\lambda^{n-1}+a_ 1\lambda^{n-2}+...+a_{n-1}=0\) lie when the coefficients satisfy the inequalities \(1\geq a_ 1\geq a_ 2\geq...\geq a_{n-1}\geq 0\) are determined. As another application, it is shown that a conjecture of \textit{N. Dmitriev} and \textit{E. Dynkin} [Izv. Akad. Nauk SSSR, Ser. Mat. 10, 167- 184 (1946)] concerning eigenvalues of stochastic matrices is false for 6 by 6 matrices, and the statement of a theorem by \textit{F. I. Karpelevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 361-383 (1951; Zbl 0043.016)] describing the region where the eigenvalues on n by n stochastic matrices lie is simplified.