  We show that for every pair of matrices (S,P), having the closed symmetrized
bidisc $\Gamma$ as a spectral set, there is a one dimensional complex algebraic
variety $\Lambda$ in $\Gamma$ such that for every matrix valued polynomial f,
the norm of f(S,P) is less then the sup norm of f on $\Lambda$.
  The variety $\Lambda$ is shown to have a particular determinantal
representation, related to the so-called "fundamental operator" of the pair
(S,P).
  When (S,P) is a strict $\Gamma$-contraction, then $\Lambda$ is a
distinguished variety in the symmetrized bidisc, i.e., a one dimensional
algebraic variety that exits the symmetrized bidisc through its distinguished
boundary. We characterize all distinguished varieties of the symmetrized bidisc
by a determinantal representation as above.
