A tighter bound for the number of words of minimum length in an automorphic orbit

Let u be a cyclic word in a free group F-n of finite rank It that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: vertical bar v vertical bar = vertical bar u vertical bar and v = 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 of degree 2n - 3 in 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 x(+/- 1) occurring in it is not equal to the total number of y(+/- 1) occurring in u. We also prove that 2n - 3 is the sharp bound for the degree of polynomials bounding N(u). As a special case, we deal with N(u) in F-2 under the same hypothesis. (c) 2006 Elsevier Inc. All rights reserved.