Uniform algebras and distinguished varieties

TL;DR

This paper explores the relationship between distinguished varieties and uniform algebras, extending results to certain polynomial images of the bidisc and analyzing polynomial convex hulls, with applications to the symmetrized bidisc.

Contribution

It establishes new connections between distinguished varieties and uniform algebras, and generalizes Samuelsson-Wold results to specific polynomial images of the bidisc.

Findings

Identifies links between distinguished varieties and uniform algebras.

Proves analogues of Samuelsson-Wold results for polynomial images of the bidisc.

Describes polynomial convex hulls of graphs of anti-holomorphic polynomials.

Abstract

In this article, we point out the connections between the distinguished varieties introduced by Agler and McCarthy with certain uniform algebras on bidisc studied by Samuelsson and Wold. We also prove analogues of Samuelsson-Wold result for the domains in $\mathbb{C}^2$ that are the images of the bidisc under certain proper polynomial map on $\mathbb{C}^2$. We also give a description of polynomial convex hull of graph of anti-holomorphic polynomial over the distinguished boundary of such domains. We mention the case for the symmetrized bidisc as an example.