Estimating heights using auxiliary functions

TL;DR

This paper extends methods for bounding algebraic number heights using auxiliary polynomials, applying these techniques to projective heights and subvarieties, and deriving new lower bounds.

Contribution

It generalizes auxiliary polynomial techniques to projective heights and subvarieties, providing a unified approach and new lower bounds for heights.

Findings

Derived lower bounds for heights on subvarieties

Unified approach for projective and subspace heights

Extended auxiliary polynomial methods to new contexts

Abstract

Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears as a solution to a certain extremal problem involving polynomials. We further generalize the above techniques to acquire both the projective height and the height on subspaces in the same way. We further obtain lower bounds on the heights of points on some subvarieties of $\mathbb P^{N-1}(\overline{\mathbb Q})$.