Existence of finite test-sets for k-power-freeness of uniform morphisms

TL;DR

This paper presents an algorithm for deciding whether uniform morphisms are k-power-free for k >= 3, demonstrating the existence of finite test-sets in this specific case, unlike the general scenario.

Contribution

The paper introduces the first known algorithm for checking k-power-freeness of uniform morphisms when k >= 3, establishing the existence of finite test-sets.

Findings

Finite test-sets exist for k-power-freeness of uniform morphisms when k >= 3

The algorithm effectively decides k-power-freeness in this case

Contrasts with the general case where such test-sets are not known to exist

Abstract

A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. We provide such an algorithm when k >= 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite test-sets for k-power-freeness.