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 Fn 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 . In this paper, we prove that N(u) is bounded by a polynomial function of degree 2n−3 in u under the hypothesis that if two letters x, y with x≠y±1 occur in u, then the total number of x±1 occurring in u 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 F2 under the same hypothesis.