Toric residue and combinatorial degree

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Consider an n-dimensional projective toric variety X defined by a convex lattice polytope P. David Cox introduced the toric residue map given by a collection of n+1 divisors Z₀,...,Zₙ on X. In the case when the Zᵢ are T-invariant divisors whose sum is X 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 P to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals I of the homogeneous coordinate ring of X. We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to I in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.