For a word equation E of length n in one variable x occurring # x times in E a resolution algorithm of O(n + # x log n) time complexity is presented here. This is the best result known and for the equations that feature # x < n log n it yields time complexity of O(n) which is optimal. Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F ∪ (uv) + u where F is a set of O(log n) words and u, v are some words such that uv is a primitive word.
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