Noncommutative rational P\'olya series

DOI10.1007/S00029-021-00629-2zbMATH Open1515.68188arXiv1906.07271MaRDI QIDQ6320651

Jason P. Bell, Daniel Smertnig

Publication date: 17 June 2019

Abstract: A (noncommutative) P'olya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^{i}mes$. We show that rational P'olya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a P'olya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

Mathematics Subject Classification ID

Valuations, completions, formal power series and related constructions (associative rings and algebras) (16W60) Algebraic theory of languages and automata (68Q70) Exponential Diophantine equations (11D61) (p)-adic and power series fields (11D88)

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