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arXiv · 1905.06450

D-finiteness, rationality, and height

Jason P. Bell·Khoa D. Nguyen·Umberto Zannier
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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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Jason P. Bell, Khoa D. Nguyen, Umberto Zannier. 2019-05-15. D-finiteness, rationality, and height. https://arxiv.org/abs/1905.06450

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