Noncommutative rational Pólya series

Abstract

A (noncommutative) Pólya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^{\times }$. We show that rational Pólya 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 Pólya 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.

35 pages; added several examples