  In this paper we study scalar multivariate subdivision schemes with general
integer expanding dilation matrix. Our main result yields simple algebraic
conditions on the symbols of such schemes that characterize their polynomial
reproduction, i.e. their capability to generate exactly the same polynomials
from which the initial data is sampled. These algebraic conditions also allow
us to determine the approximation order of the associated refinable functions
and to choose the "correct" parametrization, i.e. the grid points to which the
newly computed values are attached at each subdivision iteration. We use this
special choice of the parametrization to increase the degree of polynomial
reproduction of known subdivision schemes and to construct new schemes with
given degree of polynomial reproduction.
