Wilson fermion determinant in lattice QCD

We present a formula for reducing the rank of Wilson fermions from 4N(c)N(x)N(y)N(z)N(t) to 4N(c)N(x)N(y)N(z) keeping the value of its determinant. We analyze eigenvalues of a reduced matrix and coefficients C-n in the fugacity expansion of the fermion determinant Sigma C-n(n)(exp(mu/T))(n), which play an important role in the canonical formulation, using lattice QCD configurations on a 4(4) lattice. Numerically, log vertical bar C-n vertical bar varies as NxNyNz, and goes easily over the standard numerical range. We give a simple cure for that. The phase of Cn correlates with the distribution of the Polyakov loop in the complex plain. These results lay the groundwork for future finite density calculations in lattice QCD