Inverse polynomial images are always sets of minimal logarithmic capacity (Q312711)

Let \(f\) be an analytic function in a neighborhood of \(\infty\) and let \(\mathrm{cap}\) denote the logarithmic capacity in the complex plane. Let \(D(f,\infty)\) be the family of domains \(D\) containing \(\infty\) such that \(f\) has a single-valued analytic continuation on \(D\). A domain \(D_{0}\in D(f,\infty)\) is called extremal if \[ \mathrm{cap}(\hat{\mathbb C}\setminus D_{0})=\inf_{D\in D(f,\infty)}\mathrm{cap}(\hat{\mathbb C}\setminus D); \] in that case, \(\hat{\mathbb C}\setminus D_{0}\) is called a set of minimal logarithmic capacity with respect to \(f\). In the paper under review, the author proves that for any polynomial \(P\) there exists a function \(F\) analytic in a neighborhood of \(\infty\) such that \(P^{-1}([-1,1])\) is a set of minimal logarithmic capacity with respect to \(F\). Moreover, the relation between the functions \(P\) and \(F\) is described.