# algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$

If $K$ is a number field, let $\delta(K)$ denote the smallest positive height of an element of $K$ (recall roots of unity are precisely the points of height zero). Here is a paper where some very general bounds for $\delta(K)$ are obtained: http://arxiv.org/pdf/1203.4976v1.pdf

I'm interested in something a little different. If $L/K$ is an extension of number fields, let $\delta(L/K)$ denote the smallest positive height of an element in $L\setminus K$ (the smallest height of a "new" element). Loosely speaking, if finding $\delta(K)$ is about finding short vectors in a lattice, then finding $\delta(L/K)$ is about finding short vectors in a lattice with a sublattice removed.

Algorithms to determine the set of all elements in a number field with height less than a given bound are discussed here:

and also here

http://arxiv.org/abs/1111.4963

Clearly one could use such an algorithm to find $\delta(K)$ or $\delta(L/K)$, but might there be an easier way? Or, could there be a faster algorithm to find a new element with small height, but maybe not provably the smallest? Obviously the height of any non-torsion element of $L^\times \setminus K^\times$ is an upper bound for $\delta(L/K)$, so I guess my question is:

Is there an algorithm faster than finding the set of all points of bounded height, which might give a "good" upper bound for $\delta(L/K)$? This is a soft question, since I'm not saying what I mean by "good." The goal would be to get computational evidence to make conjectures about the behavior of $\delta(K_{i+1}/K_i)$ in certain towers of number fields $K_0 \subsetneq K_1 \subsetneq K_2 \subsetneq \cdots.$