Polynomials defining distinguished varieties

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.Comment: 26 page

Determinantal representations of semi-hyperbolic polynomials

We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.Comment: 14 pages, revisio

Rational inner functions in the Schur-Agler class of the polydisk

Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a transfer function realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.Comment: 14 page

Function theory on the Neile parabola

We give a formula for the Caratheodory distance on the Neile parabola, the variety {z^2=w^3} restricted to the bidisk; thus making it the first variety with a singularity to have its Caratheodory distance explicitly computed. In addition, we relate this to a mixed Caratheodory-Pick interpolation problem for which known interpolation theorems do not apply. Finally, we prove a bounded holomorphic function extension result from the Neile parabola to the bidisk.Comment: 19 pages. Minor error corrected in theorem 4.

The von Neumann inequality for 3x3 matrices

Recent work of Kosi\'nski on the three point Pick interpolation problem on the polydisc proves the von Neumann inequality for 3x3 matrices. We give a detailed explanation of this using several standard reductions---credit for the main result should be attributed to Kosi\'nski