  Let u be a cyclic word in a free group F_n of finite rank n that has the
minimum length over all cyclic words in its automorphic orbit, and let N(u) be
the cardinality of the set {v: |v|=|u| and v= \phi(u) for some \phi \in \text
{Aut}F_n}. In this paper, we prove that N(u) is bounded by a polynomial
function with respect to |u| under the hypothesis that if two letters x, y
occur in u, then the total number of x and x^{-1} occurring in u is not equal
to the total number of y and y^{-1} occurring in u. A complete proof without
the hypothesis would yield the polynomial time complexity of Whitehead's
algorithm for F_n.
