@article{BellSmertnig:SM:2021,
    Abstract = {A (noncommutative) P{\'o}lya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of {\$}{\$}K\^{}{$\backslash$}times {\$}{\$}. We show that rational P{\'o}lya 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{\'o}lya 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.},
    Author = {Bell, Jason and Smertnig, Daniel},
    Date = {2021/05/19},
    File = {Noncommutative rational Pólya series - s00029-021-00629-2.pdf},
    ISBN = {1420-9020},
    Journal = {Selecta Mathematica},
    Number = {3},
    Pages = {34},
    Title = {Noncommutative rational P{\'o}lya series},
    URL = {https://doi.org/10.1007/s00029-021-00629-2},
    Volume = {27},
    Year = {2021},
    bdsk-url-1 = {https://doi.org/10.1007/s00029-021-00629-2},
    date-added = {2023-06-09 15:04:24 +0200},
    date-modified = {2023-06-09 15:04:24 +0200},
    id = {Bell2021},
    doi = {10.1007/s00029-021-00629-2}
}

@article{BellSmertnig:SM:2021, Abstract = {A (noncommutative) P{\'o}lya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of {\$}{\$}K\^{}{$\backslash$}times {\$}{\$}. We show that rational P{\'o}lya 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{\'o}lya 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.}, Author = {Bell, Jason and Smertnig, Daniel}, Date = {2021/05/19}, File = {Noncommutative rational Pólya series - s00029-021-00629-2.pdf}, ISBN = {1420-9020}, Journal = {Selecta Mathematica}, Number = {3}, Pages = {34}, Title = {Noncommutative rational P{\'o}lya series}, URL = {https://doi.org/10.1007/s00029-021-00629-2}, Volume = {27}, Year = {2021}, bdsk-url-1 = {https://doi.org/10.1007/s00029-021-00629-2}, date-added = {2023-06-09 15:04:24 +0200}, date-modified = {2023-06-09 15:04:24 +0200}, id = {Bell2021}, doi = {10.1007/s00029-021-00629-2} }