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

TL;DR

This paper provides a simple proof extending Pohst's inequality to all natural numbers, establishing a universal bound on a product related to real numbers and applying it to number field discriminants.

Contribution

The authors present an elementary proof that generalizes Pohst's inequality to all degrees, impacting bounds on discriminants of totally real fields.

Findings

Proved that $P_n \,\leq\, 2^{\lfloor n/2 \rfloor}$ for all n.

Extended Pohst's inequality from n ≤ 11 to all n.

Applied the result to establish bounds on discriminants of totally real fields.

Abstract

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