Paperfolding, Automata, and Rational Functions - Diagonals and ...

Whilst (i) and (ii) are reasonably straightforward, (iii) and (iv) are more
difficult and are proved by Deligne using resolution of singularities.
Dwork has given a proof which avoids resolution.
An elementary proof of (iii) and its generalisations is given by Lipshitz:
Leonard argues so as to show D -finiteness is retained under the taking
of diagonals: roughly speaking, the vector space generated by the
partial derivatives of an algebraic power series remains finite
dimensional. I conclude by mentioning Grothendieck’s Conjecture: If a
linear homogeneous differential equation with coefficients from Q[x1] ,
and of order n , has, for almost all p , n independent solutions in
Fp[[x1]] then all its solutions are algebraic.
This has been proved in a number of special cases (for example, for
Picard-Fuchs equations) by Katz. Some results have also been
obtained by elementary methods by Honda.
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