  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.
