Uniform Approximation by Rational Functions Which All Satisfy the Same Algebraic Differential Equation

We prove that given two disjoint compact setsK1andK2in the complex plane, without any holes in them, there exists a sequencepn(z) of rational functions, all of them satisfying one and the same algebraic differential equation, such thatpn(z) converges uniformly to 1 onK1and uniformly to 0 onK2.