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

Abstract

A polygon in the complex plane is called cyclic if it is the convex hull of 1,λ, λ 2 ,… for some complex number λ. We determine the regions where λ should lie in order that this polygon be n -sided. As an application of this result we determine the region in the complex plane in which the roots of the equation λ n−1 + a 1 λ n−2 + … + a n−1 = 0 lie when the coefficients are subject to the inequalities 1 ⩾ a 1 ⩾ a 2 ⩾ … ⩾ a n−1 ⩾ 0 . As another application, we show that a conjecture of Dmitriev and Dynkin concerning eigenvalues of stochastic matrices is false for 6 by 6 matrices. We also simplify the statement of Karpelevich's theorem which describes the region M n where the eigenvalues of n by n stochastic matrices lie.