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Differential Algebra: Joseph Fels Ritt

The djvu file is a different edition - published by the AMS. The content should be essentially the same.

Review:

Published 53 years ago, this book gives an overview of an area of mathematics that has found applications in algebraic, Diophantine, and differential geometry, model theory, Painleve theory, integrable systems, automatic theorem proving, combinatorics, difference equations, and symbolic programming. It could be characterized as the study of the solutions to systems of differential equations from an algebraic point of view. These differential equations can be partial or ordinary differential equations, and can be linear or nonlinear. The book is still widely cited and can still be read profitably, despite its date of publication. The author's work was inspired by the work of the mathematicians Emily Noether and van der Waerden, and is now referred to as the Ritt theory of characteristic sets of prime differential ideals. The author's work has been developed considerably further than what he envisaged, to subfields of mathematics now known as differential algebraic geometry and differential algebraic groups.

Differential algebra can be considered to be a generalization of commutative algebra, and in chapter 1 of the book this is readily apparent. The author begins with an algebraic field of characteristic 0 and defines an operation of differentiation on it. As expected, this operation (call it ') is linear and is such that (ab)' = ba' + ab', for elememts a, b in the field. The field with the operation is then called a 'differential field', with the rational, real, and complex numbers being elementary examples. The rational and elliptic functions are also examples of differential fields. A 'differential polynomial' is then a polynomial with coefficients in a differential field. A 'differential ideal of differential polynomials' is then a collection of d.p's that is closed under linear combinations and derivatives of all orders. A 'perfect' ideal is one in which a positive integer power of one of its elements entails the element is in the ideal. A prime ideal is one in which whenever a product of is in the ideal, at least one of the factors is. In analogy with algebraic geometry, the author shows that every perfect ideal of d.p.'s has a representation as the intersection of a finite number of prime ideals.

In chapter 2, the author considers the solution set of a system of equations obtained by setting a d.p. (over indeterminates) equal to zero, this being called the 'manifold' of the d.p. A manifold is 'reducible' if it is the union of two manifolds which are proper parts; 'irreducible' otherwise. The author shows that the essential prime divisors of the perfect ideal associated to a manifold are the prime ideals associated with the components of the manifold. He then proves a 'theorem of zeros': a finite collection of d.p's having no zeros entails that some linear combination of them and their derivatives (of various orders) equals 1.

Chapter 3 outlines the structure theory of d.p's. The author gives a detailed proof of the theorem that every component of a differential polynomial of positive class in n indeterminates has dimension n - 1, as might be expected intuitively.

In chapter 4, the author turns his attention to systems of algebraic equations (which are systems of d.p's which are of order zero in each indeterminate. These systems can be understood independently of d.p's of course, but the author chooses to use the methods of earlier chapters in order to set up the tools for a study of differential equations. Polynomials and algebraic manifolds are thus the objects of interest, with an algebraic manifold being the zero set of the set of polynomials. The famous Hilbert theorom of zeros (the Nullstellensatz) is proven. The reader familiar with the theory of solutions of linear differential equations will appreciate the discussion on the construction of resolvents for a prime polynomial ideal.

The author returns to differential fields and d.p's in chapter 5, where in the first part he discusses an elimination theory for systems of algebraic differential equations. Not holding this theory to be rigorous, he develops a second one in the chapter, and compares the two. For finite systems, these considerations prove again that the manifold of solutions is the union of a finite number of irreducible algebraic differential manifolds.

The author considers the approximation theory of d.p's in chapter 6, where he proves that the general solution of an algebraically irreducible differential polynomial consists of the nonsingular zeros and of the adjacent singular zeros. The reader familiar with the Painleve theory of differential systems will see similarities to the author's discussion.

In chapter 7 the author studies the pathologies that can arise in the intersection of algebraic differential manifolds. He gives an example of an intersection at a single point, and then proves an analogue of Kronecker's theorem, namely that for a finite system of d.p's in n indeterminates, there exists a system composed of n + 1 linear combinations of the d.p's that has the same manifold as the original system.

In chapter 8, the author proves the Riquier existence theorem, which is done in order to use it in the next chapter on partial differential algebra. This theorem states that for any infinite sequence of monomials, there is one of them which is a multiple of another with index value less than its own. This theorem is used to order (analytic) functions and their partial derivatives, using a 'marking' process. These considerations motivate the concept of a 'orthonomic' system of (partial) differential equations.

In the last chapter, the author summarizes quickly what is known about partial differential algebra. Analogs of the earlier constructions are given, such as that of a partial differential polynomial. He proves that a every essential prime divisor of a nonzero partial differential polynomial has a characteristic set consisting of a single d.p. This gives a generalization of the 'spectral' theory outlined earlier for differential algebra. In addition, he gives an outline of an 'elimination theory' of algebraic partial differential equations using the operations of differentiation, rational operations, and factorizations.

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