Let $K$ be a number field with unit rank $r$, and consider the usual logarithmic map $L:K^{\ast}\rightarrow\mathbb R^r$ for which $L(\mathcal O\_K^{\times})$ is a lattice of rank $r$. Given a set $\varepsilon\_1,\ldots,\varepsilon\_r$ of fundamental units in $\mathcal O\_K$, we can consider the Euclidean lengths of the vectors $L(\varepsilon\_i)$. I would like to know a way of obtaining a suitable basis for which there is an explicit upper bound on these lengths in terms of invariants of $K$. For instance, if our basis is LLL-reduced, are there known upper bounds on the lengths of the vectors $L(\varepsilon\_i)$?



Short basis for the unit group of a number field
The geometry of numbers should say that there is a basis for the units whose lengths are bounded by an explicit function of the regulator $R\_K$. So this reduces to the question of upper bounds for $R\_K$, and of course $R\_K$ does not depend on the choice of basis. 


Since $R\_K$ is logarithmic, one might hope for a bound of the form $(\log D\_K)^t$, where $D\_K$ is the absolute discriminant, but this is almost certainly false. For example, take a real quadratic field. It is conjectured that there are infinitely many with class number 1, so the fact that $\log (h\_K R\_K) \sim \frac12\log D\_K$ says that $R\_K$ is roughly on the order of $\sqrt{D\_K}$. 


So in general, since $h\_K\ge1$, one gets $R\_K\le D\_K^{\frac12+\epsilon}$ if you range over fields with $(\log D\_K)/[K:\mathbb{Q}]\to\infty$. Presumably there are explicit and effective bounds if you're willing to accept a weaker estimate.


Finally, in the other direction, one has lower bounds of the form (constants depend on the degree of the field)
$$ R\_K \gg (\log D\_K)^{r(K)-\rho(K)}, $$
where $r(K)$ is the rank of the unit group of $K$, and $\rho(K)$ is the maximum of $r(k)$ as $k$ ranges over all proper subfields of $K$. For estimates of this sort, see


1. M. Pohst, Regulatorabschätzungen für total reelle algebraische Zahlkörper, *J. Number Theory*, **9** (1977), pp. 459–492
2. R. Remak, Über Grössenbesiehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, *Compositio Math.*, **10** (1952), pp. 245–285
3. J. Silverman, An inequality relating the regulator and the discriminant of a number field, *J. Number Theory*, **19** (1984), pp. 437–442


