Height bounds for algebraic numbers satisfying splitting conditions

TL;DR

This paper extends previous methods to establish new lower bounds on the height of algebraic numbers with conjugates confined to specific local fields, such as real intervals and p-adic discs.

Contribution

It generalizes energy minimization techniques to real and p-adic sets, providing novel height bounds for algebraic numbers with prescribed conjugate locations.

Findings

Derived new lower bounds for algebraic number heights.

Extended energy minimization methods to real intervals and p-adic discs.

Improved understanding of algebraic numbers with restricted conjugates.

Abstract

In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p-adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p-adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets.