  We study the systematic errors of L\"uscher's formulation of dynamical Wilson
quarks and some of its variants, in the weak and strong coupling limits, and on
a sample of small configurations at finite $\beta$. We confirm the existence of
an optimal window in the cutoff parameter $\varepsilon$, and the exponential
decrease of the error with the number of boson families. A non-hermitian
variant improves the approximation further and allows for an odd number of
flavors. A simple and economical Metropolis test is proposed, which makes the
algorithm exact.
