Generalization of a Pohst's inequality

Let P-n(y(1), ... y(n)) := Pi(1 <= i <= j <= n )(1 - y(i)/y(j)) and P-n := sup((y1, ..., yn) )P(n )(y(1), ..., y(n)) where the supremum is taken over the n-pies (y(1), ..., y(n)) of real numbers satisfying 0 < vertical bar y(1)< vertical bar y(2)vertical bar < ... < vertical bar y(n)&VERBAR(;). We prove that P-n <= 2(left perpendicularn/2right perpendicular) 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. (C) 2021 Elsevier Inc. All rights reserved.