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 LLLreduced, are there known upper bounds on the lengths of the vectors $L(\varepsilon_i)$?

$\begingroup$ Think about this in the 2dim case (say real quad. fields). The "short" basis will be basically consists of the first and second shortest vectors. The "bound" your asking is for how high the point representing this lattice goes to the cusp in the homogeneous space PSL2(R)/PSL2(Z). As can be seen by the basic theory of quad. forms, you can get forms of very large discriminant which will result at point very high up in the cusp (that means the shortest vector is very short, hence in the 2dim case, the second one is very long). In general I think you might get something from Baker's inequality. $\endgroup$ – Asaf May 4 '13 at 18:37

$\begingroup$ @Asaf: Aren't real quadratic fields the 1dimensional case? So your comments are probably more relevant to, say, a totally real cubic field. $\endgroup$ – Joe Silverman May 4 '13 at 21:09
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
M. Pohst, Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory, 9 (1977), pp. 459–492
R. Remak, Über Grössenbesiehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compositio Math., 10 (1952), pp. 245–285
J. Silverman, An inequality relating the regulator and the discriminant of a number field, J. Number Theory, 19 (1984), pp. 437–442