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 also briefly discuss some effects of our results on the method
of Wildanger and Smart for the resolution of S-unit equations.