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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2019
- DOI:
- 10.48550/arXiv.1906.07271
- arXiv:
- arXiv:1906.07271
- Bibcode:
- 2019arXiv190607271B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Number Theory;
- Primary 68Q45;
- 68Q70;
- Secondary 11B37
- E-Print:
- 35 pages