  We study scalar $d$-variate subdivision schemes, with dilation matrix 2I,
satisfying the sum rules of order $k$. Using the results of M\"oller and Sauer,
stated for general expanding dilation matrices, we characterize the structure
of the mask symbols of such schemes by showing that they must be linear
combinations of shifted box spline generators of some quotient polynomial
ideal. The directions of the corresponding box splines are columns of certain
unimodular matrices. The quotient ideal is determined by the given order of the
sum rules or, equivalently, by the order of the zero conditions. The results
presented in this paper open a way to a systematic study of subdivision
schemes, since box spline subdivisions turn out to be the building blocks of
any reasonable multivariate subdivision scheme. As in the univariate case, the
characterization we give is the proper way of matching the smoothness of the
box spline building blocks with the order of polynomial reproduction of the
corresponding subdivision scheme. However, due to the interaction of the
building blocks, convergence and smoothness properties may change, if several
convergent schemes are combined.
