A Detailed Proof of Pohst's Inequality

TL;DR

This paper provides a comprehensive combinatorial proof of Pohst's inequality for all dimensions, improving bounds on the regulator in totally real number fields beyond previous computational methods.

Contribution

It introduces a new combinatorial proof that extends Pohst's inequality to all n, replacing computer-assisted proofs for n ≤ 10.

Findings

Proves Pohst's inequality for all n using combinatorial methods

Provides improved bounds for the regulator in totally real number fields

Eliminates the need for computer-assisted proofs for higher dimensions

Abstract

In 1977 Pohst conjectured a certain inequality for $n$ variables and give a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms of the discriminant for totally real number fields.