Combinatorial construction of toric residues
  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. 
Amit Khetan, Ivan Soprounov