# D-finiteness, rationality, and height

## TL;DR

This paper extends the understanding of when multivariate and univariate D-finite power series are rational, using advanced number theory and combinatorial methods to analyze coefficients and their properties.

## Contribution

It generalizes previous results by allowing coefficients to form infinite sets and establishes criteria for rationality based on coefficient behavior.

## Key findings

- Multivariate D-finite series with finite coefficient sets are rational.
- Univariate D-finite series with coefficients resembling those of rational functions are rational.
- Uses advanced tools like Weil heights and the Subspace Theorem to prove rationality criteria.

## 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.