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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 and for any when the polytopes of
the divisors share a complete flag of faces. The latter generalizes earlier...
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