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

An Algebraic Equation in Characteristic p
For a prime p, and for G any formal power series with integer
coefficients,
G(X p ) ≡ (G(X)) p mod p; equivalently G(X p ) = (G(X)) p
in the ring Fp[[X]] of formal power series over the finite field Fp of p
elements. This is plain because the Frobenius map x ↦→ x p is an
additive automorphism (that is: by Fermat’s Little Theorem and
because all the multinomial coefficients other than those on the
diagonal vanish modulo p). Hence the Mahler functional equation
(1 − X 4 )F(X) 2 − (1 − X 4 )F(X) + X = 0
for F = F1 + F3 becomes the equation
(1 + X) 4 F 2 + (1 + X) 4 F + X = 0 ,
showing that F is an algebraic element over F2[[X]].
In general, a linear relation on 1, F(X), F(X p ), . . . , over Z[X]
reduces to an algebraic equation over over Fp[X] linearly relating
1, F , F p , . . . .
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