Toric residue and combinatorial degree

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 $(Z_0,\dots,Z_n)$ on $X$. In the case when the $Z_i$ are $\mathbb{T}$-invariant divisors whose sum is $X\setminus\mathbb{T}$, 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.