On the non-standard interpolations in $\mathbb{C}^n$ and combinatorial coefficients for Weil polyhedra

Multidimensional non-standard interpolation has been recently presented in an article by D. Alpay and A. Yger. We are talking about algebraic interpolation where discrete roots of a system of polynomial equations serve as nodes. With the help of the Grothendieck residue duality, the problem of describing the desired interpolation space of functions is reduced to solving the affine-bilinear equation. To implement this reduction, algorithms for calculating local Grothendieck residues or their sums are required. In a fairly general situation, the calculation of these residues is based on the well-known Gelfond–Khovanskii formula. This article provides examples of calculating local residues or their sums. In 2-dimensional case we generalise the Gelfond–Khovanskii formula for Newton polyhedra that are not in the unfolded position. This is done using the concept of an amoeba of an algebraic set and the notion of an homological resolvent for the boundary of Weil polyhedron.