The capacity of the rational preimage of a compact set
Let \(K\) be a compact set in the plane. Let \(P\) be a polynomial of degree \(n\) with leading coefficient \(1\). It is well-known that \[ \text{cap\,}(P^{-1}K)=(\text{cap\,}(K))^{1/n}, \] where \(\text{cap\,}\) is the logarithmic capacity (=transfinite diameter). In the paper under review, the author extends the above result to rational functions. Let \[ R(z)=\frac{P(z)}{Q(z)}=\frac{P(z)}{\prod_{j=1}^m(z-t_j)^{n_j}} \] be a rational function such that \(P,Q\) are irreducible polynomials with leading coefficients \(1\) for which \(\max\{\text{deg}P,\text{deg}Q\}=n\). Consider also the function \[ \phi_R(z)=\sum_{j=1}^m\frac{n_j}{n}\log|z-t_j|. \] Then the author proves the following interesting equality \[ \text{cap}_{\phi_R}(R^{-1}K)=\left (\text{cap\,}(K)\,\prod_{j=1}^m|P(t_j)|^{-n_j/n}\right )^{1/n}.\eqno{(*)} \] Here \(\text{cap}_{\phi_R}\) denotes the capacity in the external field determined by the function \(\phi_R\). The author explains in detail all the definitions and proves more general versions of the equality above.
scientific article; zbMATH DE number 6701372
scientific article