  It is well-known that polynomials decompose into spherical harmonics. This
result is called separation of variables or the Fischer decomposition. In the
paper we prove the Fischer decomposition for spinor valued polynomials in $k$
vector variables of ${\mathbb R}^m$ under the stable range condition $m\geq
2k$. Here the role of spherical harmonics is played by monogenic polynomials,
that is, polynomial solutions of the Dirac equation in $k$ vector variables.
