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Algebra and Applications #8: Difference Algebra

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Synopses & Reviews

Publisher Comments:

Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. The first stage of the development of the theory is associated with its founder J. F. Ritt (1893 - 1951) and R. Cohn whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrew the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings. This book reflects the contemporary level of difference algebra; it contains a systematic study of partial difference algebraic structures and their applications, as well as the coverage of the classical theory of ordinary difference rings and field extensions. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. It will be also of interest to researchers in computer algebra, theory of difference equations and equations of mathematical physics. The book is self-contained; it requires no prerequisites other than knowledge of basic algebraic concepts and mathematical maturity of an advanced undergraduate.

Synopsis:

This book contains a systematic study of partial difference algebraic structures and their applications, as well as coverage of the classical theory of ordinary difference rings and field extensions.

Synopsis:

Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings.

The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.

Table of Contents

Preface. 1. Preliminaries. 1.1 Basic terminology and background material. 1.2 Elements of the theory of commutative rings. 1.3 Graded and filtered rings and modules. 1.4 Numerical polynomials. 1.5 Dimension polynomials of sets of m-tuples. 1.6 Basic facts of the field theory. 1.7 Derivations and modules of differentials. 1.8 Gröbner Bases. 2. Basic concepts of difference algebra. 2.1 Difference and inversive difference rings. 2.2 Rings of difference and inversive difference polynomials. 2.3 Difference ideals. 2.4 Autoreduced sets of difference and inversive difference polynomails. Characteristic sets. 2.5 Ritt difference rings. 2.6 Varieties of difference polynomials. 3. Difference modules. 3.1 Rings of difference operators. Difference modules. 3.2 Dimension polynomials of difference modules. 3.3 Gröbner Bases with respects to several orderings and multivariable dimension polynomials of difference modules. 3.4 Inversive difference modules. 3.5 s*-Dimension polynomials and their invariants. 3.6 Dimension of general difference modules. 4. Difference field extensions. 4.1 Transformal dependence. Difference transcendental bases and difference transcendental degree. 4.2 Dimension polynomials of difference and inversive difference field extensions. 4.3 Limit degree. 4.4 The fundamental theorem on finitely generated difference field extensions. 4.5 Some results on ordinary difference field extensions. 4.6 Difference algebras. 5. Compatibility, Replicability, and Monadicity. Difference specializations. 5.1 Compatible and incompatible difference field extensions. 5.2 Difference kernels over ordinary difference fields. 5.3 Difference specializations. 5.4 Babbitt's decomposition. Criterion of compatibility. 5.5 Replicability. 5.6 Monadicity. 6. Difference kernels over partial difference fields. Difference valuation rings. 6.1 Difference kernels over partial difference fields and their prolongations. 6.2 Realizations of difference kernels over partial difference fields. 6.3 Difference valuation rings and extensions of difference specializations. 7. Systems of algebraic difference equations. 7.1 Solutions of ordinary difference polynomials. 7.2 Existence theorem for ordinary algebraic difference equations. 7.3 Existence of solutions of difference polynomials in the case of two translations. 7.4 Singular and Multiple Realizations. 7.5 Review of further results on varieties of ordinary difference polynomials. 7.6 Ritt's number. Greenspan's and Jacobi's Bounds. 7.7 Dimension polynomials and the strength of a system of algebraic difference equations. 8. Elements of the difference galois theory. 8.1 Galois correspondence for difference field extensions. 8.2 Picard-Vessiot theory of linear homogeneous difference equations. 8.3 Picard-Vessiot rings and galois theory of difference equations. Bibliography. Index.

Product Details

ISBN:
9781402069468
Author:
Levin, Alexander
Publisher:
Springer
Editor:
Fialowski, A.
Editor:
Friedlander, Eric M.
Author:
Alexander, Levin
Subject:
Algebra - General
Subject:
Differential Equations
Subject:
Difference algebra.
Subject:
Algebra - Linear
Subject:
Algebra - Abstract
Subject:
Field Theory and Polynomials
Subject:
Commutative Rings and Algebras
Subject:
Difference and Functional Equations
Subject:
Mathematics-Linear Algebra
Copyright:
Edition Description:
Book
Series:
Algebra and Applications
Series Volume:
8
Publication Date:
20080523
Binding:
HARDCOVER
Language:
English
Illustrations:
Y
Pages:
532
Dimensions:
235 x 155 mm

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Algebra and Applications #8: Difference Algebra New Hardcover
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Product details 532 pages Springer - English 9781402069468 Reviews:
"Synopsis" by , This book contains a systematic study of partial difference algebraic structures and their applications, as well as coverage of the classical theory of ordinary difference rings and field extensions.
"Synopsis" by , Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings.

The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.

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