A wealth of geometric and combinatorial properties of a given linear
endomorphism X of RN is captured in the study of its associated zonotope
Z(X), and, by duality, its associated hyperplane arrangement H(X).
This well-known line of study is particularly interesting in case n\eqbd\rank
X \ll N. We enhance this study to an algebraic level, and associate X with
three algebraic structures, referred herein as {\it external, central, and
internal.} Each algebraic structure is given in terms of a pair of homogeneous
polynomial ideals in n variables that are dual to each other: one encodes
properties of the arrangement H(X), while the other encodes by duality
properties of the zonotope Z(X). The algebraic structures are defined purely
in terms of the combinatorial structure of X, but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of Z(X) or H(X). The theory is universal in the sense that it
requires no assumptions on the map X (the only exception being that the
algebro-analytic operations on Z(X) yield sought-for results only in case X
is unimodular), and provides new tools that can be used in enumerative
combinatorics, graph theory, representation theory, polytope geometry, and
approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments
in the area since the article first appeared on the arXi