Extensions of bounded holomorphic functions on the tridisk

TL;DR

This paper investigates the structure of certain polynomially convex sets in the tridisc with extension properties, characterizing when they are retracts or graphs based on their dimension and algebraic properties.

Contribution

It provides new characterizations of polynomially convex sets in the tridisc, identifying conditions under which they are retracts or graphs, depending on their dimension and algebraic structure.

Findings

One-dimensional sets with certain properties are retracts.

Two-dimensional sets are either retracts or graphs of functions.

Conditions involving algebraic and convexity properties determine the structure.

Abstract

We study sets $V$ in the tridisc that are relatively polynomially convex and have the polynomial extension property. If $V$ is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract. If $V$ is two-dimensional, we show that either it is a retract, or, for any choice of the coordinate functions, it is the graph of a function of two variables.