Intertwining unisolvent arrays for multivariate Lagrange interpolation

Let Pd(C ) denote the space of polynomials of degree at most d in n complex variables. A subset X of C – we will usually speak of configuration or array – is said to be unisolvent for Pd(C ) (or simply unisolvent of degree d) if, for every function f defined on X there exists a unique polynomial P ∈ Pd(C ) such that P(x) = f (x) for every x ∈ X. This polynomial is called the Lagrange interpolation polynomial of f at (the points of) X and is denoted by LX[f ]. A necessary condition for X to be unisolvent of degree d is that its cardinality coincide with the dimension of Pd(C ), that is, ♯X = td(n) where td(n) := ( n+d d )