Writing det(W) in terms of an implicit parametrization of the eigenvalues is elegant, I think it's unlikely there's a better approach. Brute-forcing it would b

Writing det(W) in terms of an implicit parametrization of the eigenvalues is elegant, I think it's unlikely there's a better approach. Brute-forcing it would be to write down the multivariate polynomial in full generality, as there would be no way to break it up using the logarithm.