Differential Algebra and Related Topics

Foreword.

Workshop Participants.

Workshop Program.

Contents.

This is a tutorial article to introduce the Ritt-Kolchin theory for differential polynomials. This theory has recently been further developed and successfully applied to many computational problems where the main object of study is described by a set of algebraic differential equations. A basic knowledge of the Ritt-Kolchin theory has become a necessary prerequisite for leading-edge research in many pure as well as applied disciplines. The goal of this paper is to present the fundamental concepts and results with an emphasis on the relationship between the classical theory and the modern symbolic computation approach.

The language of schemes, which has proven to be of value to algebraic geometry, has not yet been widely accepted into differential algebraic geometry. One reason may be that there are "challenges"; some hoped-for properties of the ring of global sections are absent. In this paper we examine a class of differential rings (or modules) called AAD (Annihilators Are Differential) which meets this challenge. Any differential ring has a smallest ideal whose quotient is AAD. We use this to show that products exist in the category of AAD schemes.

Two results in Differential Algebra, Kolchin's Irreducibility Theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of "modern" or "Grothendieck style" algebraic geometry.

We survey some of the main points of contact between model theory and differential algebra. We will discuss the development of differentially closed fields, which stands at the center of the relationship between the two branches of mathematics. We will also explore recent developments in model theory of real valued function, difference algebra, and other related structures.

Let F be a differential field with field of constants C, which we assume to be algebraically closed and of characteristic 0. Let G be an algebraic group over C. The Inverse Galois Problem for G over F asks whether there is a differential Galois extension E ⊃ F whose differential Galois group G(E/F) is isomorphic to G. More generally, if E ⊃ F is a differential Galois extension and φ : G → G(E/F) is an epimorphism of algebraic groups over C, the Lifting Problem for G and E over F asks whether there is a differential Galois extension K ⊃ F containing E with G(K/F) isomorphic to G such that the natural map G(K/F) → G(E/F) becomes isomorphic to φ. (The inverse problem is the lifting problem for the case E = F.) The easiest contexts in which to understand the problems are when G is connected algebraic and, for the lifting problem, when the kernel of φ is commutative unipotent or a torus, and this tutorial will focus on those.

After a more or less standard introduction to differential Galois theory, we consider the problem of determining the universal Picard-Vessiot ring of a class of differential equations and its differential Galois group for various differential fields K and classes of equations. As a special case, we sketch a new proof of a result presented in the inspiring paper by J. Martinet and J.-P. Ramis [6].

A differential structure on an n-dimensional vector space V over a field K is a pair (δ, D), where δ is a derivation on K and D : V → V is an additive mapping satisfying the Leibniz (product) rule. A cyclic vector is a vector v such that (v, Dv,…, D^n-1v) is a basis of V. In this note we offer two existence proofs for cyclic vectors under very mild hypotheses on K. The first proof is in the form of an algorithm; the second includes a simple description of the collection of all cyclic vectors. A MAPLE implementation of the algorithm for the case is detailed in the Appendix.

We explain and illustrate how differential algebraic techniques can be used to establish the non-integrability of complex analytic Hamiltonian systems. Readers are assumed familiar with differential Galois theory, but not with Hamilton's equations; an introductory treatment of these entities, including a discussion of the significance of integrability, is therefore included. The examples we consider are handled as straightforward applications of a single result (Corollary 6.21), thereby allowing for a unified treatment of the work of several authors. Verification of the hypotheses is a matter of elementary calculation.

The purpose of this article is to explore the features of differential algebras comprised of differential invariants together with invariant differential operators. A tutorial on moving frames, used to construct complete sets of invariants and invariant differential operators, is followed by a discussion of how these sets can differ in structure from standard differential rings. We conclude with a brief description of a conjectured characteristic set algorithm for invariant over-determined differential systems.

A Baxter algebra is a commutative algebra A that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and explain more recent work on explicit constructions of free Baxter algebras that extended the constructions of Rota and Cartier. In the second part of the paper we will use these explicit constructions to relate Baxter algebras to Hopf algebras and give applications of Baxter algebras to the umbral calculus in combinatorics.