Polynomial ideals, monomial bases and a divided difference formula

Contribution to "School (and Workshop) on Computational Algebra for Algebraic Geometry and Statistics", Torino, September 2004. Summary. -A generalised (multivariate) divided difference formula is given for an arbitrary finite set of points with no subsets of three points that lie on a line. This follows from an extension of the Newton's polynomials and Newton's interpolation formula. It is derived as the interpolation based on Gröbner bases for the grid expressed as a zero-dimensional variety and is typically dependent on the chosen term-ordering and the selected ordering of points in the grid.