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.

场K上的（非交换）Pólya级数是形式幂级数，其非零系数包含在\（K ^ \ times \）的有限生成子组中。我们证明有理Pólya级数是明确的有理级数，证明了Reutenauer有40年的推测了。该证明结合了非可交换代数，自动机理论和数论（特别是单位方程）的方法。作为推论，有理数列是Pólya级数，当且仅当它是Hadamard次可逆时。用不同的措词，我们表明，在一个域的有限生成子组（和零）中采用值的每个加权有限自动机都等同于一个明确的加权有限自动机。