Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic number. A result of Schanuel asserts that $|S(K, B)| \sim c B^{2d}$ for some explicit constant $c$ depending only on the number field $K$.
Does anyone know if the proof of Schanuel's result can be turned into an algorithm for computing the set $S(K,B)$?
There are other approaches to computing the set $S(K,B)$. For example, the article "Elements with bounded height in number fields" by Petho and Schmitt [Period. Math. Hungar. 43 (2001), no. 1-2, 31–41] gives an algorithm that embeds $S(K,B)$ into a computable set of size $B^{2d^2 + d}$. But sorting through this larger set to determine $S(K,B)$ is computationally infeasible for quadratic fields when $B$ is larger than about 3.
NB - The simplest definition of the absolute multiplicative height is via the Mahler measure of its minimal polynomial. Suppose $x \in K$ has minimal polynomial $f(T) \in \mathbb{Z}[T]$, and write $$f(T)= a(T-\alpha_1) \cdots (T - \alpha_r)$$ with $\alpha_i \in \mathbb{C}$. Without loss of generality, we may assume $x = \alpha_1$. Then the absolute multiplicative height $H(x)$ can be defined by the formula
$$
H(x)^r = |a| \prod_{i = 1}^r \max\{1, |\alpha_i|\}.
$$
P.S. To clarify, I already have an algorithm in hand that calculates the set $S(K,B)$ fairly quickly, but I am specifically interested in knowing whether or not Schanuel's technique can be reworked into an algorithm.
