  We present a new algebraic algorithmic scheme to solve {\em convex integer
maximization} problems of the following form, where $c$ is a convex function on
$R^d$ and $w_1x,...,w_dx$ are linear forms on $R^n$, $$\max \{c(w_1 x,...,w_d
x): Ax=b, x\in N^n\} .$$ This method works for arbitrary input data
$A,b,d,w_1,...,w_d,c$. Moreover, for fixed $d$ and several important classes of
programs in {\em variable dimension}, we prove that our algorithm runs in {\em
polynomial time}. As a consequence, we obtain polynomial time algorithms for
various types of multi-way transportation problems, packing problems, and
partitioning problems in variable dimension.
