Criteria for regularity of Mahler power series and Becker's conjecture

TL;DR

This paper investigates the conditions under which Mahler power series are regular, proving a stronger version of Becker's conjecture for a specific subclass, thus advancing understanding of the relationship between regularity and Mahler equations.

Contribution

The paper proves a stronger form of Becker's conjecture for a subclass of Mahler power series, clarifying the criteria for regularity.

Findings

Established a stronger form of Becker's conjecture

Identified conditions linking Mahler equations and regularity

Advanced the theoretical understanding of Mahler power series

Abstract

Allouche and Shallit introduced the notion of a regular power series as a generalization of automatic sequences. Becker showed that all regular power series satisfy Mahler equations and conjectured equivalent conditions for the converse to be true. We prove a stronger form of Becker's conjecture for a subclass of Mahler power series.