自守轨道中最小长度字的计数
Counting words of minimum length in an automorphic orbit
摘要：
设u是有限秩n的自由群F_n中的循环字，且在其自守轨道上的循环字中具有最小长度，N（u）是集合{v：|v| = |u|，v= N（u），对于某个有限秩\in \text {Aut}F_n}的基数.本文证明了N（u）是由关于|u|的多项式函数有界的，条件是如果u中出现两个字母x，y，则x和x^{-1}出现在u中的总数不等于y和y^{-1}出现在u中的总数.如果没有这个假设，一个完整的证明将得到Whitehead算法的多项式时间复杂度。
英文摘要：
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= ϕ(u) for some ϕ\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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
