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^* 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.