Sur les équation fonctionelles aux itérées. (On iterated functional equations) (Q910984)

The author studies solutions of functional equations \(P(f^ 0,...,f^ s)=0\), where P is a non-zero polynomial is \(s+1\) variables and \(f^ k\) denotes the k-th iterate of the function f. It is proved that, if f is either an entire function of a complex variable or an entire function of a p-adic variable, then f is a polynomial. Moreover, it is shown that, if f is a formal power series with coefficients in a number field \({\mathbb{K}}\), then subject to some natural restrictions on f and on P f is an algebraic power series over the ring of polynomials in \(K[[x]]\).