  The toric residue is a map depending on n+1 semi-ample divisors on a complete
toric variety of dimension n. It appears in a variety of contexts such as
sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions.
In this paper we investigate the problem of finding an explicit element whose
toric residue is equal to one. Such an element is shown to exist if and only if
the associated polytopes are essential. We reduce the problem to finding a
collection of partitions of the lattice points in the polytopes satisfying a
certain combinatorial property. We use this description to solve the problem
when n=2 and for any n when the polytopes of the divisors share a complete flag
of faces. The latter generalizes earlier results when the divisors were all
ample.
