Carath\'eodory extremal functions on the symmetrized bidisc

We show how realization theory can be used to find the solutions of the Carath'eodory extremal problem on the symmetrized bidisc $$G\{(z+w,zw):\mathrm{\mid }z\mathrm{\mid }<1,\mathrm{\mid }w\mathrm{\mid }<1\}\mathrm{.}$$ We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath'eodory problems for tangents of non-generic types.