  A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex
algebraic variety that intersects $\mathbb D^n$ and exits the domain through
the $n$-torus $\mathbb T^n$ without intersecting any other part of the
topological boundary of $\mathbb D^n$. We find two different characterizations
for a distinguished variety in the polydisc $\mathbb D^n$ in terms of the
Taylor joint spectrum of certain linear matrix-pencils and thus generalize the
seminal work due to Agler and M\raise.45ex\hbox{c}Carthy [Acta Math., 2005] on
distinguished varieties in $\mathbb D^2$. We show that a distinguished variety
in $\mathbb D^n$ is a part of an affine algebraic curve which is a
set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$
is commuting tuple of Hilbert space contractions such that the defect space of
$T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a
commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$
being the minimal unitary dilation of $T$ if and only if some certain matrices
associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb
D^n$.
