Suppose $K$ is a number field, $r=r_1+r_2-1$ (the rank of the unit group) and $u_1,\dots, u_r$ are a basis of fundamental units. Suppose this basis minimizes the max over $i=1,\dots, r$ of the largest archimedean absolute value of $u_i$, and this value is $M>0$. What is a good upper bound on $M$ in terms of standard invariants of the field $K$? (In fact, I know one exists which for fixed degree is exponential in the regulator.)
More precisely, I'd like a bound on this minimax (min over choices of basis of max over basis elements) of the product of all archimedean absolute values which are greater than $1$ (a.k.a. the "entropy") of the $u_i$. (This can of course be bounded by an $r-1$st power of the previous bound.)
Is there a better bound/standard reference for bounding these quantities?