Height lower bounds for rational maps on higher-dimensional varieties

Prove strong lower bounds for h(f(x)) in terms of h(x) for rational maps of non-finite degree, particularly for rational points on higher-dimensional varieties such as abelian surfaces, in order to control the distribution of their value sets.

Background

The paper explains that a height-growth argument would yield sparsity of value sets if one could show that the height of f(x) grows rapidly relative to the height of x. Such estimates are straightforward for finite-degree maps on suitable open sets, but become substantially more difficult when fibers have positive dimension.

For rational maps from abelian surfaces, the authors obtain a conditional estimate under Vojta’s conjectures, while emphasizing that the required unconditional lower bounds are largely unavailable. Establishing such estimates is therefore an unresolved Diophantine problem underlying quantitative value-set results.