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a b s t r a c tCarl de Boor conjectured that every ideal interpolant over complex field is the pointwise limit of Lagrange interpolants. Boris Shekhtman proved that the conjecture is true in two variables, and he also provided a counterexample for more than three variables. However, Shekhtman only mentioned the existence of some mathematical objects (without giving a method to compute them) in his proof of the bivariate case. For general interpolation condition functionals on the interpolation sites, we improve Shekhtman's method to find a sequence of interpolation sites (also called discrete sites), such that the corresponding Lagrange interpolants converge to the given bivariate ideal interpolant. We discuss a special case where the multiplicity space is of breadth one. The results in this paper give a completely algorithmic way to realize Shekhtman's method.

In this paper, we study how to quickly compute the ≺-minimal monomial interpolating basis for a multivariate polynomial interpolation problem. We address the notion of "reverse" reduced basis of linearly independent polynomials and design an algorithm for it. Based on the notion, for any monomial ordering we present a new method to read off the ≺-minimal monomial interpolating basis from monomials appearing in the polynomials representing the interpolation conditions.