Efficient evaluation of multivariate polynomials over multidimensional grids

Let P(x 1 , x 2 , ... , x n ) be a multivariate polynomial in n variables, x 1 , x 2 , ... , x n and suppose that P(x 1 , x 2 , ... , x n ) is of degree m i in x i , i = 1, 2, ... , n. It is shown how P(x 1 , x 2 , ... , x n ) may be evaluated at x i = x 0i + n i Δ i for n i = 0, 1, 2, ... and i = 1, 2, ... , n using not more than (m 1 + m 2 + ... + m n ) additions per point. Feasibility of reduction from this upper bound is demonstrated in the n = 2 case.