Efficient evaluation of multivariate polynomials over multidimensional grids

Let P(x1, x2, ... , xn) be a multivariate polynomial in n variables, x1, x2, ... , xnand suppose that P(x1, x2, ... , xn) is of degree miin xi, i = 1, 2, ... , n. It is shown how P(x1, x2, ... , xn) may be evaluated at xi= x0i+ niΔifor ni= 0, 1, 2, ... and i = 1, 2, ... , n using not more than (m1+ m2+ ... + mn) additions per point. Feasibility of reduction from this upper bound is demonstrated in the n = 2 case.