The equation y′ = fy in zero residue characteristic

Abstract

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.