  We show that a particular parameter plays a vital role in the resolution of
S-unit equations, at the stage where LLL-reduction is applied. We define the
notion of optimal system of fundamental S-units (with respect to this
parameter), and prove that such a system exists and can be effectively
constructed. Applying our results and methods, one can obtain much better
bounds for the solutions of S-unit equations after the reduction step, than
earlier. We briefly also discuss some effects of our results on the method of
Wildanger and Smart for the resolution of S-unit equations.
