  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.
