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

Diagonals of Rational Functions
Now consider the class of power series in Q[[x]] which occur as the
diagonals of rational functions f (x) = P(x)/Q(x) with Q(0) = 0 .
Every algebraic power series is the diagonal of a rational power series,
and every such diagonal is algebraic mod p s for all s and almost all p.
The complete diagonals of rational power series have many other
interesting properties:
Theorem. If f (x1) is the diagonal of a rational power series over Q
(i) f has positive radius of convergence rp at every place p of Q and
rp = 1 for almost all p .
(ii) for almost all p the function f is bounded on the disc
Dp(1 − ) = {t ∈ Cp : |t| < 1} , where Cp is the completion of the
algebraic closure of the p-adic rationals Qp and, for almost all p ,
sup{|f (t)| : t ∈ Dp(1 − )} = 1 .
(iii) f satisfies a linear differential equation over Q[x1]; and
(iv) this equation is a Picard-Fuchs equation.
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