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

A Theorem of Furstenberg
It follows that the diagonal of a rational function f (x, y) of two variables
is an algebraic function. Indeed,
(If )(t) = 1
2πi
I
|y|=ɛ
f (t/y, y) dy
y
1
=
2πi
I
|y|=ɛ
P(t, y)
dy .
Q(t, y)
Writing Q(t, y) = Q` y − yi(t) ´ , where the yi(t) are algebraic, and
evaluating the integral by residues verifies the claim.
In fact, every algebraic power series of one variable is the diagonal of a
rational power series of two variables and, indeed, every algebraic
power series in n variables is the diagonal of a rational power series in
2n variables.
Below I sketch a proof showing that every diagonal of a rational
function defined over a finite field is algebraic. However, in
characteristic zero diagonals of rational functions in more than two
variables do not generally yield algebraic functions.
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