Quadratic word equations with regular constraints and the exponent of periodicity

2025/11/11 by Volker Diekert, Silas Natterer, Diekert, Volker +3 · 1 voice
Computer Science · Mathematics · #Commutative property #Conjecture #Exponent #Formal Methods in Verification #Geometric and Algebraic Topology #Quadratic equation #Variety (cybernetics) #Word (group theory) #Word problem (mathematics education) #cs.FL #math.RA #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2511.08208

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

Abstract

In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in the broader framework of word equations with regular constraints and establish new positive results: the conjecture holds for all quadratic word equations with constraints in finite semigroups from the variety DLG and its left-right dual DRG, encompassing, in particular, all finite groups, commutative semigroups, and J-trivial semigroups.

Discussions

• Volker Diekert, Silas Natterer, Alexander Thumm: Word equations and the exponent of periodicity https://arxiv.org/abs/2511.08208 https://arxiv.org/pdf/2511.08208 https://arxiv.org/html/2511.08208 [bsky, 0 points, 0 comments]

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