The Weil height in terms of an auxiliary polynomial

TL;DR

This paper introduces a new theorem that uses auxiliary polynomials to establish lower bounds on the Weil height of any algebraic number, generalizing previous results and providing new bounds.

Contribution

The paper presents a novel theorem that extends auxiliary polynomial methods to all algebraic numbers, broadening the scope of height lower bounds.

Findings

Generalizes previous bounds on Weil height using auxiliary polynomials

Provides new lower bounds in specific algebraic cases

Contains corollaries that extend earlier theorems

Abstract

Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $\alpha$ under certain assumptions on $\alpha$. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. Our theorem contains, as corollaries, a slight generalization of the above results as well as some new lower bounds in other special cases.