### A New Formula for the Volume of Lattice Polyhedra.

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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 $n=2$ and for any $n$ when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier...

We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry ${D}_{4}$, and the Gosset polytope ${4}_{21}$ with the symmetry ${E}_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope ${4}_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the ${E}_{8}$ or ${D}_{4}$ actions on the ${4}_{21}$ or the $24$-cell, respectively. Moreover, we show that each...