Maximal ratio of coefficients of divisors and an upper bound for height for rational maps

TL;DR

This paper introduces the D-ratio for rational maps to establish an upper bound for height functions, extending the known bounds from morphisms to rational maps in algebraic geometry.

Contribution

It defines the D-ratio of a rational map, providing a new tool to bound height functions where traditional degree-based bounds fail.

Findings

D-ratio effectively replaces degree in height inequalities for rational maps.

The new bounds improve understanding of height growth under rational maps.

The approach generalizes height bounds from morphisms to rational maps.

Abstract

When we have a morphism f : P^n -> P^n, then we have an inequality \frac{1}{\deg f} h(f(P)) +C > h(P) which provides a good upper bound of $h(P)$. However, if $f$ is a rational map, then \frac{1}{\deg f} h(f(P))+C cannot be an upper bound of h(P). In this paper, we will define the $D$-ratio of a rational map $f$ which will replace the degree of a morphism in the height inequality of h(P).