An elementary proof for a generalization of a Pohst's inequality

Abstract

Let $P_n(y_1,\dots ,y_n):={\prod }_{1\le i<j\le n}\left(1-\frac{y_i}{y_j}\right)$ and $P_n:={\sup }_{(y_1,\dots ,y_n)}{P}_n(y_1,\dots ,y_n)$ where the supremum is taken over the $n$-ples $(y_1,\dots ,y_n)$ of real numbers satisfying $0<|y_1|<|y_2|<\cdots <|y_n|$. We prove that $P_n\le 2^{\lfloor n/2\rfloor }$ for every $n$, i.e., we extend to all $n$ the bound that Pohst proved for $n\le 11$. 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.

10 pages