Cauchy-Weil formula, Schur-Agler type classes, new Hardy spaces of the polydisk and interpolation problems

Given n rational inner functions Φ=(Φ₁,…,Φ_n) defining a 0-dimensional (hence finite) complete intersection in the polydisk Dⁿ and a Weil polyhedron Δ_r^Φ relatively compact in Dⁿ, we combine together the Cauchy-Weil representation formulas respectively in Dⁿ (considered as a Weil polyhedron subordinated to the coordinate functions) and Δ_r^Φ in order to provide integral representations formulas involving a positive kernel provided Φ satisfies some Schur-Agler type conditions. We then study interpolation problems in H₂(Dⁿ) along the finite set Φ⁻¹({0_}). One of the main objectives of this paper is to emphasize the important role played here by Cauchy-Weil representation formula (as in computational polynomial geometry), together with its intimate connection with multivariate residue calculus.