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We consider languages expressed by word equations in two variables and give a complete
characterization for their complexity functions, that is, the functions that give the number of
words of the same length. Specifically, we prove that there are only five types of complexities:
constant, linear, exponential, and two in between constant and linear. For the latter two, we
give precise characterizations in terms of the number of solutions of Diophantine equations of
certain types. In particular,...
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