Rational Functions, Diagonals, Automata and Arithmetic

1.1 By R we denote an integral domain; then R[[x]] is the ring of formal power series over R in the variables x = (x1, . . . , xn). We write a series f(x) ∈ R[[x]] as f(x) = ∑ ai1···inx i1 1 · · ·xin n = ∑ aνx ν where ν = (ν1, . . . , νn) is a multi-index and x = x1 1 · · ·xνn n . In the sequel R is usually the finite field Fp of p elements or the ring of p-adic integers Zp. One says that f(x) ∈ R[[x]] is algebraic if it is algebraic over the quotient field of the polynomial ring R[x]. If f(x) ∈ Zp[[x]] we say that f is algebraic mod p if there is an algebraic g(x) ∈ Zp[[x]] with f ≡ g mod p.