A fast algorithm for the multivariate Birkhoff interpolation problem

Multivariate Birkhoff interpolation is the most complicated polynomial interpolation problem and the theory about it is far from systematic and complete. In this paper we derive an Algorithm B-MB (Birkhoff-Monomial Basis) and prove B-MB giving the minimal interpolation monomial basis w.r.t. the lexicographical order of the multivariate Birkhoff problem. This algorithm is the generalization of Algorithm MB in [L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73-87] which is a well known fast algorithm used to compute the interpolation monomial basis of the Hermite interpolation problem.