Applications of branched values to p-adic functional equations on analytic functions

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form g ^q = hf ^q + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.