Algebraic zeros divisors on the projective line having small diagonals and small heights and their application to adelic dynamics

TL;DR

This paper proves a quantitative adelic equidistribution theorem for algebraic zeros divisors with small diagonals and heights on the projective line, and applies it to adelic dynamics of rational functions over product formula fields.

Contribution

It introduces a new quantitative adelic equidistribution theorem for algebraic zeros divisors with small diagonals and heights, extending to arbitrary characteristic and non-separable cases.

Findings

Established a quantitative adelic equidistribution theorem.

Derived local proximity estimates for iterations of rational functions.

Applied results to adelic dynamics of rational functions.

Abstract

We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to an adelic normalized weight $g$ in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function $f\in k(z)$ of degree $>1$ and a rational function $a\in k(z)$ of degree $>0$ over a product formula field $k$ of characteristic $0$, applying this quantitative adelic equidistribution result to adelic dynamics of $f$.