  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 $n >
3$.
