Difference fields and descent in algebraic dynamics, II

TL;DR

This paper advances the theory of difference fields in algebraic dynamics by extending descent techniques to more general settings, enabling a sharper formulation of the dynamical Northcott theorem.

Contribution

It generalizes descent theory to rational maps, arbitrary base fields, and correspondences, providing a detailed decomposition of difference field extensions.

Findings

Decomposition of difference field extensions into finite, internal, and one-based parts

Strengthened descent theory applicable to broader dynamical systems

Precise formulation of the dynamical Northcott theorem

Abstract

This second part of the paper strengthens the descent theory described in the first part to rational maps, arbitrary base fields, and dynamics given by correspondences. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the "dynamical Northcott" Theorem 1.11 of Part I in sharp form.