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Mapping words to powers by morphisms

Aleksi Saarela

Abstract

We characterize the words that can be mapped to arbitrarily high powers by injective morphisms. For all other words, we prove a linear upper bound for the highest power that they can be mapped to, and this bound is optimal up to a constant factor if there is no restriction on the size of the alphabet. We also prove that, for any integer $n \geq 2$, deciding whether a given word can be mapped to an $n$th power by a nonperiodic morphism is NP-hard and in PSPACE, and so is deciding whether a given word can be mapped to a nonprimitive word by a nonperiodic morphism.

Mapping words to powers by morphisms

Abstract

We characterize the words that can be mapped to arbitrarily high powers by injective morphisms. For all other words, we prove a linear upper bound for the highest power that they can be mapped to, and this bound is optimal up to a constant factor if there is no restriction on the size of the alphabet. We also prove that, for any integer , deciding whether a given word can be mapped to an th power by a nonperiodic morphism is NP-hard and in PSPACE, and so is deciding whether a given word can be mapped to a nonprimitive word by a nonperiodic morphism.

Paper Structure

This paper contains 6 sections, 21 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results
  4. High powers
  5. Algorithmic questions
  6. Conclusion

Key Result

Theorem 2.2

Let $\Sigma = \{x, y, z\}$ and $k, m, n \geq 2$. The constant-free word equation $(x^k, y^m z^n)$ has only periodic solutions.

Theorems & Definitions (43)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Example 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 33 more