Abstract

Word equations of the form x^k = z₁^k₁z₂^k₂...z_n^k_n are considered in this paper. In particular, we investigate the case where x is of different length than z_i, for any i, and k and k_i are at least 3, for all 1 ≤ i ≤ n, and n ≤ k. We prove that for those equations all solutions are of rank 1, that is, x and z_i are powers of the same word for all 1 ≤ i ≤ n. It is also shown that this result implies a well-known result by K.I. Appel and F.M. Djorup about the more special case where k_i = k_j for all 1 ≤ i < j ≤ n.

Keywords: combinatorics on words, word equations

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