Generalized Pohst inequality and small regulators

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 a 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 ( r 1 , r 2 ) = ( 6 , 1 ) (r_1,r_2)=(6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures ( 3 , 1 ) (3,1) , ( 4 , 1 ) (4,1) and ( 5 , 1 ) (5,1) .