Uniform approximation by rational functions which all satisfy the same algebraic differential equation (Q1908216)

If \(K_1\), \(K_2\) are compact sets in \(\mathbb{C}\), with connected complements and \(K_1 \cap K_2 = \emptyset\), then there exists a sequence \(\{r_n\}\) of rational functions such that \(r_n \to 0\) on \(K_1\), \(r_n \to 1\) on \(K_2\), and such that these \(r_n\) satisfy one and the same algebraic differential equation. It is not known whether there are polynomials with these properties.