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 McCarthy [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 \(\mathbb D^n\) .
