Distribution of zeros of the Hermite-Pad, polynomials for a system of three functions, and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Pad, polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface that has a canonical decomposition. We consider a system of three functions that are rational on the constructed Riemann surface and satisfy the independence condition det [f(k)(z((j)))] not equivalent to 0. In the case of m = 3, we refine the main theorem from Nuttall's paper of 1981. In particular, we show that in this case the complement a",I" \ B of the open (possibly, disconnected) set B aS, a",I" introduced in Nuttall's paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Pad, approximants.