Counting words of minimum length in an automorphic orbit

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 {nu: vertical bar v vertical bar = vertical bar u vertical bar and nu = phi (u) for some phi is an element of Aut F-n}. In this paper, we prove that N (u) is bounded by a polynomial function with respect to vertical bar u vertical bar under the hypothesis that if two letters x, y with x not equal y(+/- 1) occur in u, then the total number of occurrences of x(+/- 1) in u is not equal to the total number of occurrences of y(+/- 1) in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for F-n. (c) 2006 Elsevier Inc. All rights reserved.