# Noncommutative rational P\'olya series

## TL;DR

This paper proves that rational noncommutative Pólya series are unambiguous, confirming a long-standing conjecture, and characterizes them via Hadamard sub-invertibility, linking automata theory and algebra.

## Contribution

It proves the conjecture that rational Pólya series are unambiguous and establishes their equivalence to automata with finitely generated subgroup weights.

## Key findings

- Rational Pólya series are unambiguous rational series.
- A rational series is a Pólya series iff it is Hadamard sub-invertible.
- Weighted automata with subgroup weights are equivalent to unambiguous automata.

## 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 $K^\times$. 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.