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

So we have:
Theorem. P aνx ν ∈ Fp[[x]] is algebraic if and only if it is generated by
a finite automaton.
Corollary. And so: Let f , g ∈ Fp[[x]] be algebraic. Then
(i) Every diagonal of f is algebraic, as is every off-diagonal.
(ii) The Hadamard product f ∗ g is algebraic.
(iii) Irrelevance of symbols: Each characteristic series f (i) = P
is algebraic.
aν=i x ν
We have already noted that the situation in characteristic zero is very
different.
Indeed, in characteristic zero there is a theorem of Polyá –Carleson by
which a power series with integer coefficients and radius of
convergence 1 is either rational or has the unit circle as a natural
boundary. Thus the Mahler functions mentioned here all are
transcendental functions.
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