Generalization of a Pohst's inequality

Let [Formula presented] and Pn:=sup(y1,…,yn)⁡Pn(y1,…,yn) where the supremum is taken over the n-ples (y1,…,yn) of real numbers satisfying 0<|y1|<|y2|<⋯<|yn|. We prove that Pn≤2⌊n/2⌋ for every n, i.e., we extend to all n the bound that Pohst proved for n≤11. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.