n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

Let $D$ be a bounded Jordan domain and $A$ be its complement on the Riemann sphere. We investigate the $n$-th root asymptotic behavior in $D$ of best rational approximants, in the uniform norm on $A$, to functions holomorphic on $A$ having a multi-valued continuation to quasi every point of $D$ with finitely many branches. More precisely, we study weak${}^{\ast }$ convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $n$-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.