Extremal holomorphic maps and the symmetrized bidisc

We introduce the class of n-extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Gamma. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence C-nu, where nu >= 0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that Cn-3 is insufficient for the solvability of an n-point problem for n >= 3. We propose the conjecture that condition Cn-2 is necessary and sufficient for the solvability of an n-point interpolation problem for Gamma and we explore the implications of this conjecture. We introduce a classification of rational Gamma-inner functions, that is, analytic functions from the disc into Gamma whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Gamma. The classes are related to n-extremality and the conditions C-nu; we prove numerous strict inclusions between the classes.