Pure inner functions, distinguished varieties and toral algebraic commutative contractive pairs

. We study the pairs of commuting contractions that are annihilated by polynomials with a geometric condition on its zero set, herein called the toral algebraic pairs. Toral algebraic pairs of commuting isometries are characterized. In particular, a commuting pair of isometries is toral algebraic if and only if so is its minimal unitary extension. This triggers the natural question when a toral algebraic pair of commuting contractions lifts, in the sense of Ando, to a toral algebraic pair of commuting isometries. While this question remains open, a family including all the commuting contractive matrices is obtained for which the answer is affirmative. The study involves understanding of certain matrix-valued analytic functions, which in turn, throws new light on certain algebraic varieties which are studied extensively from operator- and function-theoretic point of view over the last two decades - the so-called distinguished varieties.