  Criterion for a companion matrix to have a certain number of flat portions on
the boundary of its numerical range is given. The criterion is specialized to
the cases of 3-by-3 and 4-by-4 matrices. In the latter case, it is proved that
a 4-by-4 unitarily irreducible companion matrix cannot have 3 flat portions on
the boundary of its numerical range. Numerical examples are given to illustrate
the main results.
