Asymptotic uniqueness of best rational approximants of given degree to Markov functions in L-2 of the circle

We improve over a sufficient condition given in [8] for uniqueness of a non degenerate critical point in best rational approximation of prescribed degr ee over the conjugate-symmetric Hardy space (H) over bar (2)(R) of the comp lement of the disk. The improved condition connects to error estimates in A AK approximation, and is necessary and sufficient when the function to be a pproximated is of Markov type. For Markov functions whose defining measure satisfies the Szego condition, we combine what precedes with sharp asymptot ics in multipoint Pade approximation from [43], [40] in order to prove uniq ueness of a critical point when the degree of the approximant goes large. T his lends perspective to the uniqueness issue for more general classes of f unctions defined through Cauchy integrals.