Using a sums of squares formula for two variable polynomials with no zeros on
the bidisk, we are able to give a new proof of a representation for
distinguished varieties. For distinguished varieties with no singularities on
the two-torus, we are able to provide extra details about the representation
formula and use this to prove a bounded extension theorem.

A certain kernel (sometimes called the Pick kernel) associated to Schur
functions on the disk is always positive semi-definite. A generalization of
this fact is well-known for Schur functions on the polydisk. In this article,
we show that the Pick kernel on the polydisk has a great deal of structure
beyond being positive semi-definite. It can always be split into two kernels
possessing certain shift invariance properties.

We prove a detailed sums of squares formula for two variable polynomials with
no zeros on the bidisk $\mathbb{D}^2$ extending previous versions of such a
formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to
the Christoffel-Darboux formula for orthogonal polynomials on the unit circle,
but the extension to two variables involves issues of uniqueness in the formula
and the study of ideals of two variable orthogonal polynomials with respect to
a positive Borel measure on the torus which may have infinite mass.