  We introduce the class of $n$-extremal holomorphic maps, a class that
generalises 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 symmetrised 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 $\mathcal{C}_\nu, \nu \geq 0,$ of necessary conditions for
solvability, prove that they are of strictly increasing strength and show that
$\mathcal{C}_{n-3}$ is insufficient for the solvability of an $n$-point problem
for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}_{n-2}$ 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
$\mathcal{C}_\nu$; we prove numerous strict inclusions between the classes.

Complex Variables