D-finiteness, rationality, and height
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- by Jason P. Bell, Khoa D. Nguyen and Umberto Zannier PDF
- Trans. Amer. Math. Soc. 373 (2020), 4889-4906 Request permission
Abstract:
Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic $0$ is D-finite and its coefficients belong to a finite set, then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series âlook likeâ the coefficients of a rational function, then the power series is rational. Our work relies on the theory of Weil heights, the ManinâMumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.References
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Additional Information
- Jason P. Bell
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 632303
- Email: jpbell@uwaterloo.ca
- Khoa D. Nguyen
- Affiliation: Department of Mathematics and Statistics, University of Calgary, AB T2N 1N4, Canada
- MR Author ID: 886774
- Email: dangkhoa.nguyen@ucalgary.ca
- Umberto Zannier
- Affiliation: Scuola Normale Superiore, Classe di Scienze Matematiche e Naturali, Pisa, Italy
- MR Author ID: 186540
- Email: umberto.zannier@sns.it
- Received by editor(s): June 5, 2019
- Received by editor(s) in revised form: October 8, 2019, October 9, 2019, and October 28, 2019
- Published electronically: April 29, 2020
- Additional Notes: The first author was partially supported by an NSERC Discovery Grant.
The second author was partially supported by a start-up grant at the University of Calgary and an NSERC Discovery Grant. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4889-4906
- MSC (2010): Primary 11D61, 11G50; Secondary 13F25
- DOI: https://doi.org/10.1090/tran/8046
- MathSciNet review: 4127865