Abstract: The minimal degree Newton basis for Lagrange interpolation on a lower subset of a tensor product grid that is proposed by Gasca and Sauer was extended to a tower subset of the grid. At first, we solved the bivariate cases. Furthermore, we extended the notion of a 2dimensional tower set to arbitrary high dimensionsand then gave the minimal degree Newton basis for trivariate Lagrange interpolation on a tower set as an example. Finally, we introduced a fast algorithm for constructing the reduced Grbner basis for the vanishingideal of a 3dimensional tower set.

Key words: tower set, multivariate polynomial interpolation, minimal degree Newton basis