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Full-Text Articles in Physical Sciences and Mathematics
Toric Residue And Combinatorial Degree, Ivan Soprunov
Toric Residue And Combinatorial Degree, Ivan Soprunov
Mathematics and Statistics Faculty Publications
Consider an -dimensional projective toric variety defined by a convex lattice polytope . David Cox introduced the toric residue map given by a collection of divisors on . In the case when the are -invariant divisors whose sum is , the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals of the homogeneous coordinate ring of . We give a necessary and sufficient …
Combinatorial Construction Of Toric Residues, Amit Khetan, Ivan Soprunov
Combinatorial Construction Of Toric Residues, Amit Khetan, Ivan Soprunov
Mathematics and Statistics Faculty Publications
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