Closing the gap around the essential minimum of height functions with linear programming

TL;DR

This paper demonstrates that two classical methods for bounding the essential minimum of height functions are dual in linear programming, establishing strong duality and enabling precise computation and realization of the minimum.

Contribution

It proves the strong duality between lower and upper bound methods for the essential minimum, closing the gap and enabling new computational approaches.

Findings

Strong duality between the two classical methods

The essential minimum can be realized by a generic sequence of algebraic integers

If the Green function is computable, then the essential minimum is a computable real number

Abstract

For many common height functions, it is notoriously hard to compute the essential minimum. Nevertheless there are two classical methods, one giving lower bounds and the other giving upper bounds. In this paper, we show that the two methods are actually dual to each other in the sense of linear programming. The main theorem is that they satisfy strong duality, which closes the gap around the essential minimum from both ends. As applications we prove that this essential minimum can be realized by a generic sequence of algebraic integers, and that if the associated Green function is computable then this essential minimum is a computable real number.