A polynomial shared by certain differential polynomials (Q2685728)

Let \(f\) and \(g\) be two nonconstant meromorphic functions in the complex plane \(\mathbb{C}\). If \(a\in\mathbb{C}\cup\infty\), then we denote by \(\overline{E}(a;f)\) the set of zeros of \(f-a\), and by \(E(a;f)\) we denote the set of pairs \(z,\nu\) such that \(z\) is a zero of \(f-a\) with multiplicity \(\nu\) (here the poles of \(f\) are regarded as zeros of \(f-\infty\)). The functions \(f\) and \(g\) share the value \(a\) CM (counting multiplicities) if \(E(a;f)=E(a;g)\), and they share the value \(a\) IM (ignoring multiplicities) if \(\overline{E}(a;f)=\overline{E}(a;g)\). The authors study the uniqueness problem for meromorphic functions sharing a differential polynomial.