Quantitative height bounds under splitting conditions

TL;DR

This paper refines lower bounds for the height of algebraic numbers satisfying splitting conditions by explicitly incorporating degree dependence using discrete energy methods on the Berkovich projective line.

Contribution

It introduces a new approach with discrete energy techniques to make degree dependence explicit in height bounds under splitting conditions.

Findings

Lower bounds depend only on local properties of algebraic numbers.

Explicit degree dependence in height bounds is established.

Improved bounds over previous work for algebraic numbers with splitting conditions.

Abstract

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being totally real or p-adic, improving on earlier work of Bombieri and Zannier in the totally p-adic case. These bounds applied as the degree of the algebraic number over the rationals tended towards infinity. In this paper, we use discrete energy approximation techniques on the Berkovich projective line to make the dependence on the degree in these bounds explicit, and we establish lower bounds for algebraic numbers which depend only on local properties of the numbers.