Demystifying the Karpelevic theorem

TL;DR

This paper simplifies the understanding of the Karpelevic theorem by proving the existence and properties of specific arcs on the boundary of the eigenvalue region of stochastic matrices, advancing the proof of the theorem.

Contribution

It proves the existence of continuous arcs satisfying Ito's polynomial equations and shows these arcs are simple, contributing to the proof of the Karpelevic theorem.

Findings

Existence of continuous functions tracing boundary arcs

Arcs are proven to be simple curves

Boundary points are extremal for matrix size n > 3

Abstract

The statement of the Karpelevic theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevic region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito -- in particular, Ito's theorem asserts that the boundary of the Karpelevic region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent. More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevic region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence the Karpelevic theorem. The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function $\lambda:[0,1] \longrightarrow \mathbb{C}$ such that $\mathsf{P}^\mathsf{I}(\lambda(\alpha)) = 0$, $\forall \alpha \in [0,1]$, where $\mathsf{P}^\mathsf{I}$ is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevic region are extremal whenever $n > 3$.