  A stationary subdivision scheme generates the full space of polynomials of
degree up to $k$ if and only if its mask satisfies sum rules of order $k+1$, or
its symbol satisfies zero conditions of order $k+1$. This property is often
called the polynomial reproduction property of the subdivision scheme. It is a
well-known fact that this property is, in general, only necessary for the
associated refinable function to have approximation order $k+1$.
  In this paper we study a different polynomial reproduction property of
multivariate scalar subdivision scheme with dilation matrix $mI$, $|m| \ge 2$.
Namely, we are interested in capability of a subdivision scheme to reproduce in
the limit exactly the same polynomials from which the data is sampled. The
motivation for this paper are the results by Adi Levin that state that such a
reproduction property of degree $k$ of the subdivision scheme is sufficient for
having approximation order $k+1$.
  Our main result yields simple algebraic conditions on the subdivision symbol
for computing the exact degree of such polynomial reproduction and also for
determining the associated parametrization. The parametrization determines the
grid points to which the newly computed values are attached at each subdivision
iteration to ensure the higher degree of polynomial reproduction. We illustrate
our results with several examples.
