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We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size… (More)

In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z + ∪ {+∞}. Let k ∈ Z + ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A * are said to be k-Abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the number of… (More)

We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on… (More)

A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities. Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3.

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ∈ N1∪{+∞} where N1 denotes the set of positive integers. Two finite words u and v in A * are said to be k-abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the… (More)

It is known that there are recurrent words with constant abelian complexity three, but not with constant complexity four. We prove that there are recurrent words with ultimately constant complexity c for every c.

Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length k and, moreover, start and end with a same factor of length k − 1, respectively. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze… (More)

We analyze and reprove the famous theorem of Hmelevskii, which states that the general solutions of constant-free equations on three unknowns are finitely parameterizable, that is expressible by a finite collection of formulas of word and numerical parameters. The proof is written, and simplified, by using modern tools of combinatorics on words. As a new… (More)

We consider with a new point of view the notion of nth powers in connection with the k-abelian equivalence of words. For a fixed natural number k, words u and v are k-abelian equivalent if every factor of length at most k occurs in u as many times as in v. The usual abelian equivalence coincides with 1-abelian equivalence. Usually k-abelian squares are… (More)