Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the L_p convergence of a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2-norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.