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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

D-finiteness, rationality, and height


Authors: Jason P. Bell, Khoa D. Nguyen and Umberto Zannier
Journal: Trans. Amer. Math. Soc. 373 (2020), 4889-4906
MSC (2010): Primary 11D61, 11G50; Secondary 13F25
DOI: https://doi.org/10.1090/tran/8046
Published electronically: April 29, 2020
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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 ``look like'' the coefficients of a rational function, then the power series is rational. Our work relies on the theory of Weil heights, the Manin-Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.


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Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: jpbell@uwaterloo.ca

Khoa D. Nguyen
Affiliation: Department of Mathematics and Statistics, University of Calgary, AB T2N 1N4, Canada
Email: dangkhoa.nguyen@ucalgary.ca

Umberto Zannier
Affiliation: Scuola Normale Superiore, Classe di Scienze Matematiche e Naturali, Pisa, Italy
Email: umberto.zannier@sns.it

DOI: https://doi.org/10.1090/tran/8046
Keywords: D-finite power series, Weil height, polynomial-exponential equations
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.
Article copyright: © Copyright 2020 American Mathematical Society