On an arithmetic inequality on $\mathbb{P}^1_{\mathbb{Q}}$

TL;DR

This paper proves an inequality relating height and $\chi$-arithmetic volume for toric metrized divisors on $ ext{P}^1_{ ext{Q}}$, addressing a question in arithmetic geometry.

Contribution

It establishes a new inequality connecting height and $\chi$-arithmetic volume for toric divisors, partially answering a question posed by Burgos, Moriwaki, Philippon, and Sombra.

Findings

Established an inequality between height and $\chi$-arithmetic volume.

Provides partial resolution to a question in arithmetic geometry.

Advances understanding of toric metrized divisors on $ ext{P}^1_{ ext{Q}}$.

Abstract

We establish an inequality comparing the height and the $\chi$-arithmetic volume of toric metrized divisors on $\mathbb{P}^1_{\mathbb{Q}}$. This gives a partial answer to a question of Burgos, Moriwaki, Philippon and Sombra ([5, remark 5.13]).