  Masser and Vaaler have given an asymptotic formula for the number of
algebraic numbers of given degree $d$ and increasing height. This problem was
solved by counting lattice points (which correspond to minimal polynomials over
$\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The
volume of this star body was computed by Chern and Vaaler, who also computed
the volume of the codimension-one "slice" corresponding to monic polynomials --
this led to results of Barroero on counting algebraic integers. We show how to
estimate the volume of higher-codimension slices, which allows us to count
units, algebraic integers of given norm, trace, norm and trace, and more. We
also refine the lattice point-counting arguments of Chern-Vaaler to obtain
explicit error terms with better power savings, which lead to explicit versions
of some results of Masser-Vaaler and Barroero.
