Finite Interpolation with Minimum Uniform Norm in Cn

Given a finite sequence a≔{a1, …, aN} in a domain Ω⊂Cn, and complex scalars v≔{v1, …, vN}, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying f(aj)=vj for all j. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough so that its A(Ω)-hull contains a subset of the original a large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which contains the hull (in an appropriate, weaker, sense) of a measure supported on the maximum modulus set. An example is given to show that the inclusions can be strict.