  Current methods for the classification of number fields with small regulator
depend mainly on an upper bound for the discriminant, which can be improved by
looking for the best possible upper bound of a specific polynomial function
over an hypercube. In this paper, we provide new and effective upper bounds for
the case of fields with one complex embedding and degree between five and nine:
this is done by adapting the strategy we have adopted to study the totally real
case, but for this new setting several new computational issues had to be
overcome. As a consequence, we detect the four number fields of signature (6,1)
with smallest regulator; we also expand current lists of number fields with
small regulator in signatures (3,1), (4,1) and (5,1).
