Finitely generated extensions of difference fields

Weak Presentations of Non-Finitely Generated Fields

Let K be a countable field. Then a weak presentation of K is an isomorphism of K onto a field whose elements are natural numbers, such that all the field operations are extendible to total recursive functions. Given a pair of two non-finitely generated countable fields contained in some overfield, we investigate under what circumstances the overfield has a weak presentation under which the give...

Decomposable extensions of difference fields

We define the decomposable extensions of difference fields and study the irreducibility of q-Painlevé equation of type A 7 ′ . Every strongly normal extension or Liouville-Franke extension, the latter of which is a difference analogue of the Liouvillian extension, satisfies that its appropriate algebraic closure is a decomposable extension.

Arithmetic Height Functions over Finitely Generated Fields

In this paper, we propose a new height function for a variety defined over a finitely generated field overQ. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). CONTENTS Introduction 1 1. Arakelov intersection theory 3 2. Arithmetically positive hermitian line bundles 6 3. Ari...

The Kolchin Seminar in Differential Algebra Finitely Generated Difference Field Extensions

A characterization of finitely generated multiplication modules

 Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...