Well-defined determinant representations for non-normal multivariate rational interpolants

If the system of linear equations defining a multivariate rational interpolant is singular, then the table of multivariate rational interpolants displays a structure where the basic building block is a hexagon. Remember that for univariate rational interpolation the structure is built by joining squares. In this paper we associate with every entry of the table of rational interpolants a well-defined determinant representation, also when this entry has a nonunique solution. These determinant formulas are crucial if one wants to develop a recursive computation scheme.In section 2 we repeat the determinant representation for nondegenerate solutions (nonsingular systems of interpolation conditions). In theorem 1 this is generalized to an isolated hexagon in the table. In theorem 2 the existence of such a determinant formula is proven for each entry in the table. We conclude with an example in section 5.