Finite interpolation with minimum uniform norm in C-n

Given a finite sequence a := {a(1), ..., a(N)} in a domain Omega subset of C-n, and complex scalars r:= {r(1), ..., r(N)}, consider the classical extr emal problem of finding the smallest uniform norm of a holomorphic function integral(a(t)) = r(integral) for all integral. We show that the modulus of the solutions to this problem must approach its least upper hound along a subset of the boundary of the domain large enough KI that its A(Omega)-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 contain s 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 b e strict, (C) 2000 Academic Press.