  We consider the algebraic K-theory of a truncated polynomial algebra in
several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This
naturally leads to a new generalization of the big Witt vectors. If k is a
perfect field of positive characteristic we describe the K-theory computation
in terms of a cube of these Witt vectors on N^n. If the characteristic of k
does not divide any of the a_i we compute the K-groups explicitly. We also
compute the K-groups modulo torsion for k=Z. To understand this K-theory
spectrum we use the cyclotomic trace map to topological cyclic homology, and
write TC(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)) as the iterated homotopy
cofiber of an n-cube of spectra, each of which is easier to understand.
  Updated: This is a substantial revision. We corrected several errors in the
description of the Witt vectors on a truncation set on N^n and modified the key
proofs accordingly. We also replaces several topological statement with purely
algebraic ones. Most arguments have been reworked and streamlined.
