Polynomials defining distinguished varieties

TL;DR

This paper presents a new proof for representing distinguished varieties using sums of squares formulas and extends the understanding of their structure, including a bounded extension theorem for certain cases.

Contribution

It introduces a novel proof technique for distinguished varieties and provides detailed representations and extension results for varieties without singularities on the two-torus.

Findings

New sum of squares representation for distinguished varieties

Extended representation details for non-singular varieties on the two-torus

Proved a bounded extension theorem for specific distinguished varieties

Abstract

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.