# The polynomial cluster value problem

## TL;DR

This paper investigates the polynomial cluster value problem in complex Banach spaces, establishing new theorems for uniform algebras and providing insights into the original cluster value problem for specific space classes.

## Contribution

It introduces several polynomial cluster value theorems for uniform algebras and extends results to the original cluster value problem for particular Banach spaces.

## Key findings

- Proved polynomial cluster value theorems for $H(B)$ algebras.
- Derived new results for the original cluster value problem.
- Analyzed spaces like continuous functions, $ ext{l}_1$, and locally uniformly convex spaces.

## Abstract

The polynomial cluster value problem replaces the role of the continuous linear functionals in the original cluster value problem for the continuous polynomials to describe the corresponding cluster sets and fibers. We prove several polynomial cluster value theorems for uniform algebras $H(B)$ between $A_u(B)$ and $H^{\infty}(B)$, where $B$ is the open unit ball of a complex Banach space $X$. We also obtain new results about the original cluster value problem, especially for $A_{\infty}(B)$. Examples of spaces $X$ considered here are spaces of continuous functions, $\ell_1$ and locally uniformly convex spaces.